%------------------------------------------------------------------------------
% File     : ITP270_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DeleteBounds 00283_015243
% 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    : 0074_VEBT_DeleteBounds_00283_015243 [Des22]

% Status   : Theorem
% Rating   : 0.44 v8.2.0, 0.60 v8.1.0
% Syntax   : Number of formulae    : 13294 (4179 unt;2438 typ;   0 def)
%            Number of atoms       : 27543 (10611 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19673 (2986   ~; 529   |;1892   &)
%                                         (2724 <=>;11542  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of types       :  444 ( 443 usr)
%            Number of type conns  : 2319 (1487   >; 832   *;   0   +;   0  <<)
%            Number of predicates  :  108 ( 105 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1890 (1890 usr; 508 con; 0-8 aty)
%            Number of variables   : 25928 (25047   !; 881   ?;25928   :)
% 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 12:10:16.675
%------------------------------------------------------------------------------
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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__Num__Onum_J,type,
    fun_int_num: $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__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $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__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
    filter_int: $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__List__Olist_It__HOL__Obool_J,type,
    list_bool: $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__Product____Type__Ounit,type,
    product_unit: $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__String__Oliteral,type,
    literal: $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 (1995)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_option_nat > fun_option_nat_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_real ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_int_int ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_nat_nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_nat_nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_nat_int > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_int_int > fun_int_complex ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_int ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_real ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_int_int ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: extended_enat > fun_fu263853427677366787at_rat ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: extended_enat > fun_fu263853427677366787at_rat ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_nat_rat > fun_fu8328862064278946247at_rat ).

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

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_Ex1451620480498224165t_real > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_Ex4152069839811072081at_rat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_Extended_enat > fun_fu739766798173130341t_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_Ex1451620480498224165t_real > fun_fu4696490002264930058t_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_complex_rat > fun_fu4947971316162376536ex_rat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_Ex4152069839811072081at_rat > fun_fu8357369079510409782at_rat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_real_nat > fun_fu4405974721632067664al_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: set_Extended_enat > fun_fu3738011059562840620t_bool ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_Extended_enat > fun_fu5527893311006036234t_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_int > fun_fu337402001079966300t_bool ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_int > fun_fu163127709195666036t_bool ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_complex > fun_fu3810065756161129304x_bool ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_complex > fun_fu9216119399439645550x_bool ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_Extended_enat > fun_fu8394451710700313048t_bool ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_Extended_enat > fun_fu5057839232988439434t_bool ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_rat > fun_nat_rat ).

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

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

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

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: nat > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: nat > fun_co1378794544173401837x_bool ).

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

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_option_nat_nat > fun_fu8387170092758243159at_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_option_nat_real > fun_fu8047470260750595855t_real ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_op5246294523634652747omplex > fun_fu805630362171458961t_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_set_nat_complex > fun_fu1617006900395023197t_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_option_nat_real > fun_option_nat_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_set_nat_real > fun_set_nat_real ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_Pr672294278902479040t_real > fun_Pr672294278902479040t_real ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_Ex1451620480498224165t_real > fun_Ex1451620480498224165t_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_real_bool ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_real_bool ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: int > fun_int_int ).

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_bool ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_int_nat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_real_real ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_real_real ).

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

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_real > fun_fu6899955116716589935t_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_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: real > fun_re4238438383274954274t_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: 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_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_real > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_real_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_real_bool > fun_re1255401033344146673l_bool ).

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

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_real_real > fun_fu3077872570260440615l_real ).

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

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_real_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_real_bool ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_real_bool ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_nat_rat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: rat > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: extended_enat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: extended_enat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: real > fun_re8658731967611918585t_real ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: set_Extended_enat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_na819858167508000762st_nat ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_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: fun_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_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: fun_rat_rat ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re3324582151780938536l_bool: ( fun_in5519481650651148416r_bool * fun_bo1841576699660343469l_bool ) > fun_fu2005298298577998622l_bool ).

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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__Int__Oint_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__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: list_set_nat > fun_nat_set_nat ).

tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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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,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
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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__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oreplicate_001t__Extended____Nat__Oenat,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Option__Ooption_It__Nat__Onat_J,type,
    replicate_option_nat: ( nat * option_nat ) > list_option_nat ).

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
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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__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Nat__Bijection_Olist__decode,type,
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tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
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tff(sy_c_Nat__Bijection_Olist__encode,type,
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tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__decode,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__encode,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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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,
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tff(sy_c_Num_Onat__of__num,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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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__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onum__of__nat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Opow,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Num_Osqr,type,
    sqr: fun_num_num ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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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__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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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_OLeast_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,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__Rat__Orat,type,
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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__Extended____Nat__Oenat_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__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__Filter__Ofilter_It__Nat__Onat_J,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__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__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,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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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,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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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,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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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,
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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,
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tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
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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__Int__Oint,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__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
    set_or5403411693681687835d_enat: ( extended_enat * extended_enat ) > set_Extended_enat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

tff(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: ( literal * fun_Pr1792556611952799308t_real ) > real ).

tff(sy_c_String_OLiteral,type,
    literal2: ( bool * bool * bool * bool * bool * bool * bool * literal ) > literal ).

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_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_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: fun_nat_real > $o ).

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_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

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_Opowr__real,type,
    powr_real2: 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__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bi_tot896582865486249351at_int: fun_Pr5932607294251420486t_bool > $o ).

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

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

tff(sy_c_Typedef_Otype__definition_001t__Extended____Nat__Oenat_001t__Option__Ooption_It__Nat__Onat_J,type,
    type_d7649664348572268762on_nat: ( fun_Ex621978932285409945on_nat * fun_op7883857814936610463d_enat * set_option_nat ) > $o ).

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: vEBT_VEBT > 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,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: 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,type,
    vEBT_T_m_i_n_t: 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__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 * vEBT_VEBT ) > vEBT_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_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr5442483421567724333t_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_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: vEBT_VEBT > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_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__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__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_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun_nat_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,
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    aa_fun9137951878280015556l_bool: ( fun_fu7307090382573344987l_bool * fun_real_real ) > bool ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_fun9137271176455094897l_bool: ( fun_fu7376312551830202376l_bool * fun_real_real ) > fun_real_bool ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun6946558556880649704l_real: ( fun_fu3077872570260440615l_real * fun_real_real ) > fun_real_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
    aa_fun334843040499603765l_bool: ( fun_fu2568719150003003596l_bool * fun_real_real ) > fun_fu7376312551830202376l_bool ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Complex__Ocomplex_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J,type,
    aa_fun5767801665285160478t_real: ( fun_fu1617006900395023197t_real * fun_set_nat_complex ) > fun_set_nat_real ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_fun3386010016858254564at_nat: ( fun_fu1590586450949517859at_nat * fun_set_nat_nat ) > fun_set_nat_nat ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J,type,
    aa_fun3569254276315353500t_real: ( fun_fu5354524256550068955t_real * fun_set_nat_real ) > fun_set_nat_real ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__HOL__Obool_J_001t__fun_It__fun_It__Real__Oreal_Mt__HOL__Obool_J_Mt__HOL__Obool_J,type,
    aa_fun7221085148252759141l_bool: ( fun_fu8602805464270671924l_bool * fun_fun_nat_rat_bool ) > fun_fu7511176317168184086l_bool ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__HOL__Obool,type,
    aa_fun1696419067750587730t_bool: ( fun_fu8308374331699588969t_bool * fun_fu8328862064278946247at_rat ) > bool ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun8686829112900759002l_real: ( fun_fu3711856879280950809l_real * fun_fu8328862064278946247at_rat ) > fun_real_real ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__HOL__Obool_J,type,
    aa_fun9159459243913713430l_bool: ( fun_fu762719556336274221l_bool * fun_fu8328862064278946247at_rat ) > fun_fu7307090382573344987l_bool ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_Mt__HOL__Obool_J,type,
    aa_fun4901480949278917796t_bool: ( fun_fu4669108322068655547t_bool * fun_fu8328862064278946247at_rat ) > fun_fu8308374331699588969t_bool ).

tff(sy_c_fAll_001t__VEBT____Definitions__OVEBT,type,
    fAll_VEBT_VEBT: fun_VEBT_VEBT_bool > bool ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_bool > real ).

tff(sy_c_fEx_001t__Nat__Onat,type,
    fEx_nat: fun_fu7168253649841883962l_bool ).

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: complex > fun_complex_bool ).

tff(sy_c_fequal_001t__Extended____Nat__Oenat,type,
    fequal_Extended_enat: extended_enat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: fun_bo1841576699660343469l_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Nat__Onat_J,type,
    fequal_option_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1841576699660343469l_bool ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > fun_set_complex_bool ).

tff(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > fun_se2015444840911768724t_bool ).

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__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $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__Option__Ooption_It__Nat__Onat_J,type,
    member_option_nat: ( option_nat * set_option_nat ) > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: ( produc3843707927480180839at_nat * set_Pr4329608150637261639at_nat ) > $o ).

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____Definitions__OVEBT,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_x,type,
    x: nat ).

% Relevant facts (10785)
tff(fact_0_False,axiom,
    x != mi ).

% False
tff(fact_1__C5_Ohyps_C_I8_J,axiom,
    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,aa_num_num(bit0,one))),deg))) ).

% "5.hyps"(8)
tff(fact_2__092_060open_062mi_A_060_Ax_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),x)) ).

% \<open>mi < x\<close>
tff(fact_3_less__exp,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,aa_num_num(bit0,one))),N))) ).

% less_exp
tff(fact_4_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_5_semiring__norm_I83_J,axiom,
    ! [N: num] : one != aa_num_num(bit0,N) ).

% semiring_norm(83)
tff(fact_6_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_7_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_8_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_9_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_10__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% \<open>2 \<le> deg\<close>
tff(fact_11_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_12_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_13_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_14_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_15_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,M) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_16_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_17_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_18_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_19_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_20__C5_Ohyps_C_I7_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "5.hyps"(7)
tff(fact_21__092_060open_062_092_060not_062_A_Ix_A_061_Ami_A_092_060and_062_Ax_A_061_Ama_J_092_060close_062,axiom,
    ~ ( ( x = mi )
      & ( x = ma ) ) ).

% \<open>\<not> (x = mi \<and> x = ma)\<close>
tff(fact_22_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))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X)) ) ) ) ).

% max_in_set_def
tff(fact_23_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))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X3)) ) ) ) ).

% min_in_set_def
tff(fact_24__092_060open_062x_A_092_060le_062_Ama_A_092_060and_062_Ami_A_092_060le_062_Ax_092_060close_062,axiom,
    ( 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,mi),x)) ) ).

% \<open>x \<le> ma \<and> mi \<le> x\<close>
tff(fact_25_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_26_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_27_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_28_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_29_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(78)
tff(fact_30_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_31_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_32_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_33_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_34_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_35_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_36_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_37_verit__la__disequality,axiom,
    ! [A: rat,B: 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,B),A)) ) ).

% verit_la_disequality
tff(fact_38_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ 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,B),A)) ) ).

% verit_la_disequality
tff(fact_39_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ 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,B),A)) ) ).

% verit_la_disequality
tff(fact_40_verit__la__disequality,axiom,
    ! [A: int,B: 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,B),A)) ) ).

% verit_la_disequality
tff(fact_41_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_42_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_43_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_44_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_45_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_46_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,aa_num_num(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_47_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_48_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,aa_num_num(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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_57_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_58_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_59_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_60_order__refl,axiom,
    ! [X: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X),X)) ).

% order_refl
tff(fact_61_order__refl,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).

% order_refl
tff(fact_62_order__refl,axiom,
    ! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).

% order_refl
tff(fact_63_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_64_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_65_dual__order_Orefl,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% dual_order.refl
tff(fact_66_dual__order_Orefl,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% dual_order.refl
tff(fact_67_dual__order_Orefl,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% dual_order.refl
tff(fact_68_dual__order_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% dual_order.refl
tff(fact_69_dual__order_Orefl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% dual_order.refl
tff(fact_70_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_71_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,aa_num_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_72_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(aa_num_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_73_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_74_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_75_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,aa_num_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_76_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(aa_num_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_77_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_78_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_79_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,aa_num_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_80_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(aa_num_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_81_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_82_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_83_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,aa_num_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_84_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(aa_num_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_85_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_86_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_87_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
         => pp(aa_nat_bool(P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
           => ( ! [I2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I2))
                 => pp(aa_nat_bool(P,I2)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_88_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_89_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_90_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_91_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_92_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_93_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_94_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(71)
tff(fact_95_mem__Collect__eq,axiom,
    ! [A: option_nat,P: fun_option_nat_bool] :
      ( member_option_nat(A,collect_option_nat(P))
    <=> pp(aa_option_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_96_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_97_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_98_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_99_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_100_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_101_Collect__mem__eq,axiom,
    ! [A3: set_option_nat] : collect_option_nat(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_102_Collect__mem__eq,axiom,
    ! [A3: set_real] : collect_real(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_103_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_104_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A3)) = A3 ).

% Collect_mem_eq
tff(fact_105_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_106_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_ae(A3)) = A3 ).

% Collect_mem_eq
tff(fact_107_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(Q,X4)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_108_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
        <=> pp(aa_list_nat_bool(Q,X4)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_109_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
        <=> pp(aa_set_nat_bool(Q,X4)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_110_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
        <=> pp(aa_nat_bool(Q,X4)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_111_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(Q,X4)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_112_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_113_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).

% of_nat_numeral
tff(fact_114_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = aa_num_real(numeral_numeral_real,N) ).

% of_nat_numeral
tff(fact_115_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% of_nat_numeral
tff(fact_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_124_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_125_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_126_of__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) ).

% of_nat_power
tff(fact_127_of__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) ).

% of_nat_power
tff(fact_128_of__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) ).

% of_nat_power
tff(fact_129_of__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) ).

% of_nat_power
tff(fact_130_of__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) ).

% of_nat_power
tff(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_142_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_143_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_144_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_145_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,aa_num_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_146_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_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_147_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_148_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_149_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_150_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,aa_num_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_151_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(aa_num_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(3)
tff(fact_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_184_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_185_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_186_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_187_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(pow,X),one) = X ).

% pow.simps(1)
tff(fact_188_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_189_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_190_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_191_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_192_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_193_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ 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_eq_rat,Y),X)) ) ).

% linorder_le_cases
tff(fact_194_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_le_cases
tff(fact_195_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_196_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_197_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_198_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_199_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_200_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_201_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_202_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_203_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_204_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_205_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_206_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_207_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_208_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_209_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_210_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_211_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_212_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_213_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_214_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_215_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_216_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_217_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( 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_eq_rat,Y),X)) ) ).

% linorder_linear
tff(fact_218_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_linear
tff(fact_219_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_220_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_221_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% order_eq_refl
tff(fact_222_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_eq_refl
tff(fact_223_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_eq_refl
tff(fact_224_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_225_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_226_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_227_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_228_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_229_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_230_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_231_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( 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,aa_num_num(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_232_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_233_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_234_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_235_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_236_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_237_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_238_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_239_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_240_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_241_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_242_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_243_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_244_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_245_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_246_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_247_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: 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,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_248_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( 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,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_249_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( 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,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_250_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: 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,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_251_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_252_antisym,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,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_253_antisym,axiom,
    ! [A: num,B: num] :
      ( 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,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_254_antisym,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_eq_nat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_255_antisym,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,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_256_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A)) ) ) ).

% dual_order.trans
tff(fact_257_dual__order_Otrans,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),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% dual_order.trans
tff(fact_258_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( 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),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% dual_order.trans
tff(fact_259_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_260_dual__order_Otrans,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),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% dual_order.trans
tff(fact_261_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_262_dual__order_Oantisym,axiom,
    ! [B: rat,A: 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,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_263_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( 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,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_264_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( 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,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_265_dual__order_Oantisym,axiom,
    ! [B: int,A: 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,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_266_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_267_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( 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,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_268_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( 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,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_269_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = 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_eq_nat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_270_dual__order_Oeq__iff,axiom,
    ! [A: int,B: 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_eq_int,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_271_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A4: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A4),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A4),B3)) )
     => ( ! [A4: rat,B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A4),B3)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_272_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A4),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A4),B3)) )
     => ( ! [A4: num,B3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A4))
           => pp(aa_num_bool(aa_num_fun_num_bool(P,A4),B3)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_273_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A4),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B3)) )
     => ( ! [A4: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_274_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A4),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A4),B3)) )
     => ( ! [A4: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A4))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A4),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_275_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Z)) ) ) ).

% order_trans
tff(fact_276_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( 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_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) ) ) ).

% order_trans
tff(fact_277_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) ) ) ).

% order_trans
tff(fact_278_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% order_trans
tff(fact_279_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),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,X),Z)) ) ) ).

% order_trans
tff(fact_280_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).

% order.trans
tff(fact_281_order_Otrans,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,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% order.trans
tff(fact_282_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( 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,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% order.trans
tff(fact_283_order_Otrans,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,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% order.trans
tff(fact_284_order_Otrans,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,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% order.trans
tff(fact_285_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_286_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( 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_eq_rat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_287_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_288_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_289_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_290_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( ( B = C )
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_291_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_292_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_293_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_294_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_295_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_296_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( 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,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_297_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_298_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( 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,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_299_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( 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,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_300_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_301_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = 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_eq_rat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_302_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_303_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_304_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_305_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( 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_eq_rat,Y),Z)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_306_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_307_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_308_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),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,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( 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,Z),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_309_nle__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,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_310_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ 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,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_311_nle__le,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_eq_nat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_312_nle__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,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_313_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_314_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X)) ).

% lt_ex
tff(fact_315_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X)) ).

% lt_ex
tff(fact_316_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_1)) ).

% gt_ex
tff(fact_317_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_1)) ).

% gt_ex
tff(fact_318_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_1)) ).

% gt_ex
tff(fact_319_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_1)) ).

% gt_ex
tff(fact_320_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y)) ) ) ).

% dense
tff(fact_321_dense,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ? [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y)) ) ) ).

% dense
tff(fact_322_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_323_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_324_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_325_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_326_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_327_order_Oasym,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,B),A)) ) ).

% order.asym
tff(fact_328_order_Oasym,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,B),A)) ) ).

% order.asym
tff(fact_329_order_Oasym,axiom,
    ! [A: num,B: 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,B),A)) ) ).

% order.asym
tff(fact_330_order_Oasym,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,B),A)) ) ).

% order.asym
tff(fact_331_order_Oasym,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,B),A)) ) ).

% order.asym
tff(fact_332_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_333_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( 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,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_334_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_335_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( 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,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_336_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( 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,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_337_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_338_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_339_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_340_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_341_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_342_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X4))
             => pp(aa_nat_bool(P,Y4)) )
         => pp(aa_nat_bool(P,X4)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_343_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_344_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_345_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_346_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_347_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_348_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_349_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ( X != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_cases
tff(fact_350_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_351_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_352_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_353_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% dual_order.asym
tff(fact_354_dual__order_Oasym,axiom,
    ! [B: rat,A: 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,A),B)) ) ).

% dual_order.asym
tff(fact_355_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( 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,A),B)) ) ).

% dual_order.asym
tff(fact_356_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( 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,A),B)) ) ).

% dual_order.asym
tff(fact_357_dual__order_Oasym,axiom,
    ! [B: int,A: 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,A),B)) ) ).

% dual_order.asym
tff(fact_358_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_359_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% dual_order.irrefl
tff(fact_360_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% dual_order.irrefl
tff(fact_361_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_362_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_363_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ? [N3: nat] :
          ( pp(aa_nat_bool(P,N3))
          & ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
             => ~ pp(aa_nat_bool(P,M3)) ) ) ) ).

% exists_least_iff
tff(fact_364_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B: real] :
      ( ! [A4: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A4),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B3)) )
     => ( ! [A4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),A4))
       => ( ! [A4: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_365_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A4: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A4),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A4),B3)) )
     => ( ! [A4: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A4),A4))
       => ( ! [A4: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A4),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_366_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A4),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A4),B3)) )
     => ( ! [A4: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A4),A4))
       => ( ! [A4: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A4))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A4),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_367_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B3)) )
     => ( ! [A4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),A4))
       => ( ! [A4: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_368_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A4),B3)) )
     => ( ! [A4: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A4),A4))
       => ( ! [A4: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A4))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A4),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_369_order_Ostrict__trans,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,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_370_order_Ostrict__trans,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,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans
tff(fact_371_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: 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,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans
tff(fact_372_order_Ostrict__trans,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,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_373_order_Ostrict__trans,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,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_374_not__less__iff__gr__or__eq,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,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_375_not__less__iff__gr__or__eq,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,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_376_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_377_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ 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))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_378_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ 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))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_379_dual__order_Ostrict__trans,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),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_380_dual__order_Ostrict__trans,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),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_381_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( 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),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_382_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_383_dual__order_Ostrict__trans,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),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_384_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_385_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_386_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_387_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_388_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_389_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_390_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_391_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_392_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_393_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_394_linorder__neqE,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
tff(fact_395_linorder__neqE,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
tff(fact_396_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_397_linorder__neqE,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
tff(fact_398_linorder__neqE,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
tff(fact_399_order__less__asym,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,Y),X)) ) ).

% order_less_asym
tff(fact_400_order__less__asym,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,Y),X)) ) ).

% order_less_asym
tff(fact_401_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_402_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( 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)) ) ).

% order_less_asym
tff(fact_403_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( 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)) ) ).

% order_less_asym
tff(fact_404_linorder__neq__iff,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_neq_iff
tff(fact_405_linorder__neq__iff,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_neq_iff
tff(fact_406_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_407_linorder__neq__iff,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_neq_iff
tff(fact_408_linorder__neq__iff,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_neq_iff
tff(fact_409_order__less__asym_H,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,B),A)) ) ).

% order_less_asym'
tff(fact_410_order__less__asym_H,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,B),A)) ) ).

% order_less_asym'
tff(fact_411_order__less__asym_H,axiom,
    ! [A: num,B: 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,B),A)) ) ).

% order_less_asym'
tff(fact_412_order__less__asym_H,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,B),A)) ) ).

% order_less_asym'
tff(fact_413_order__less__asym_H,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,B),A)) ) ).

% order_less_asym'
tff(fact_414_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_trans
tff(fact_415_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: 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,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_trans
tff(fact_416_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_trans
tff(fact_417_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( 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),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_trans
tff(fact_418_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( 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),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_trans
tff(fact_419_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_420_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_421_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_422_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_423_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_424_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_425_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_426_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_427_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_428_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_429_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_430_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_431_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_432_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_433_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_434_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_435_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_436_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_437_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_438_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_439_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_440_order__less__irrefl,axiom,
    ! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).

% order_less_irrefl
tff(fact_441_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_442_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_443_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_444_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_445_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_446_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_447_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_448_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_449_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_450_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_451_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_452_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_453_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_454_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_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(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_455_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_456_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_457_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_458_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_459_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_460_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_461_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_462_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_463_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_464_order__less__not__sym,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,Y),X)) ) ).

% order_less_not_sym
tff(fact_465_order__less__not__sym,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,Y),X)) ) ).

% order_less_not_sym
tff(fact_466_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_467_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( 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)) ) ).

% order_less_not_sym
tff(fact_468_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( 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)) ) ).

% order_less_not_sym
tff(fact_469_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_470_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: bool] :
      ( 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))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_471_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_472_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( 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))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_473_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( 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))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_474_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_475_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      | ( X = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_less_linear
tff(fact_476_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_477_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_478_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_479_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_480_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_481_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_482_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_483_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_484_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_485_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_486_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_487_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_488_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_489_order__less__imp__not__less,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,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_490_order__less__imp__not__less,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,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_491_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_492_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( 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)) ) ).

% order_less_imp_not_less
tff(fact_493_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( 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)) ) ).

% order_less_imp_not_less
tff(fact_494_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_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_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)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_503_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_504_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_505_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_506_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_507_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_508_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_509_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y)) ) ).

% leD
tff(fact_510_leD,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% leD
tff(fact_511_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_512_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_513_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_514_leI,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_eq_real,Y),X)) ) ).

% leI
tff(fact_515_leI,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_eq_rat,Y),X)) ) ).

% leI
tff(fact_516_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_517_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ 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_eq_nat,Y),X)) ) ).

% leI
tff(fact_518_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ 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_eq_int,Y),X)) ) ).

% leI
tff(fact_519_nless__le,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_eq_real,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_520_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_521_nless__le,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_eq_rat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_522_nless__le,axiom,
    ! [A: num,B: 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_eq_num,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_523_nless__le,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,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_524_nless__le,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,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_525_antisym__conv1,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_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_526_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_527_antisym__conv1,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_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_528_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_529_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ 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_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_530_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ 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_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_531_antisym__conv2,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 ) ) ) ).

% antisym_conv2
tff(fact_532_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_533_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( 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,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_534_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_535_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_536_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_537_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_ge
tff(fact_538_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_ge
tff(fact_539_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_le
tff(fact_540_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Z)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_le
tff(fact_541_less__le__not__le,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_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_542_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).

% less_le_not_le
tff(fact_543_less__le__not__le,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_eq_rat,X),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_544_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_545_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( 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_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_546_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( 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_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_547_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_548_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% not_le_imp_less
tff(fact_549_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_550_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_551_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_552_order_Oorder__iff__strict,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_real,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_553_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_554_order_Oorder__iff__strict,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_rat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_555_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( 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_num,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_556_order_Oorder__iff__strict,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,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_557_order_Oorder__iff__strict,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_int,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_558_order_Ostrict__iff__order,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_eq_real,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_559_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_560_order_Ostrict__iff__order,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_eq_rat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_561_order_Ostrict__iff__order,axiom,
    ! [A: num,B: 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_eq_num,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_562_order_Ostrict__iff__order,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,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_563_order_Ostrict__iff__order,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,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_564_order_Ostrict__trans1,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_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_565_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).

% order.strict_trans1
tff(fact_566_order_Ostrict__trans1,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_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_567_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( 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_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans1
tff(fact_568_order_Ostrict__trans1,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_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_569_order_Ostrict__trans1,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_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_570_order_Ostrict__trans2,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_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_571_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).

% order.strict_trans2
tff(fact_572_order_Ostrict__trans2,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_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_573_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: 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_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans2
tff(fact_574_order_Ostrict__trans2,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,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_575_order_Ostrict__trans2,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_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_576_order_Ostrict__iff__not,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_eq_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_577_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).

% order.strict_iff_not
tff(fact_578_order_Ostrict__iff__not,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_eq_rat,A),B))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_579_order_Ostrict__iff__not,axiom,
    ! [A: num,B: 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_eq_num,A),B))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_580_order_Ostrict__iff__not,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,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_581_order_Ostrict__iff__not,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,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_582_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
     => ( ! [W2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W2)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_583_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
     => ( ! [W2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),X))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W2)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_584_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_585_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ! [W2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Y))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_586_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_587_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_588_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: 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_rat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_589_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( 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_num,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_590_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( 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,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_591_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: 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_int,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_592_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_593_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_594_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: 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_eq_rat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_595_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( 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_eq_num,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_596_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( 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_eq_nat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_597_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: 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_eq_int,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_598_dual__order_Ostrict__trans1,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_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_599_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_600_dual__order_Ostrict__trans1,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_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_601_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( 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_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_602_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_603_dual__order_Ostrict__trans1,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_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_604_dual__order_Ostrict__trans2,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_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_605_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_606_dual__order_Ostrict__trans2,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_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_607_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( 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_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_608_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_609_dual__order_Ostrict__trans2,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_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_610_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_611_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_612_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: 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_eq_rat,B),A))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_613_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( 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_eq_num,B),A))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_614_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( 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_eq_nat,B),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_615_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: 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_eq_int,B),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_616_order_Ostrict__implies__order,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_eq_real,A),B)) ) ).

% order.strict_implies_order
tff(fact_617_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% order.strict_implies_order
tff(fact_618_order_Ostrict__implies__order,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_eq_rat,A),B)) ) ).

% order.strict_implies_order
tff(fact_619_order_Ostrict__implies__order,axiom,
    ! [A: num,B: 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_eq_num,A),B)) ) ).

% order.strict_implies_order
tff(fact_620_order_Ostrict__implies__order,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,A),B)) ) ).

% order.strict_implies_order
tff(fact_621_order_Ostrict__implies__order,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,A),B)) ) ).

% order.strict_implies_order
tff(fact_622_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_623_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_624_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: 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_eq_rat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_625_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( 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_eq_num,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_626_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( 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_eq_nat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_627_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: 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_eq_int,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_628_order__le__less,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 ) ) ) ).

% order_le_less
tff(fact_629_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_630_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( 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,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_631_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_632_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_633_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_634_order__less__le,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_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_635_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_636_order__less__le,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_eq_rat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_637_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_638_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( 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_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_639_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( 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_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_640_linorder__not__le,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,Y),X)) ) ).

% linorder_not_le
tff(fact_641_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ 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,Y),X)) ) ).

% linorder_not_le
tff(fact_642_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_643_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_644_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_645_linorder__not__less,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_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_646_linorder__not__less,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_eq_rat,Y),X)) ) ).

% linorder_not_less
tff(fact_647_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_648_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ 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_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_649_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ 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_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_650_order__less__imp__le,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_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_651_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% order_less_imp_le
tff(fact_652_order__less__imp__le,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_eq_rat,X),Y)) ) ).

% order_less_imp_le
tff(fact_653_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_654_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( 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_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_655_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( 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_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_656_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ( A != B )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_657_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( ( A != B )
       => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).

% order_le_neq_trans
tff(fact_658_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( A != B )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_659_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( A != B )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_660_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( A != B )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_661_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( A != B )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_662_order__neq__le__trans,axiom,
    ! [A: real,B: 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_real,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_663_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).

% order_neq_le_trans
tff(fact_664_order__neq__le__trans,axiom,
    ! [A: rat,B: 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_rat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_665_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( 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_num,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_666_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( 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,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_667_order__neq__le__trans,axiom,
    ! [A: int,B: 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_int,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_668_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_669_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(Y),Z))
       => pp(aa_set_nat_bool(ord_less_set_nat(X),Z)) ) ) ).

% order_le_less_trans
tff(fact_670_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( 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,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_671_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_672_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( 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,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_673_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),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,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_674_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_675_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z))
       => pp(aa_set_nat_bool(ord_less_set_nat(X),Z)) ) ) ).

% order_less_le_trans
tff(fact_676_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: 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_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_677_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_678_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( 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_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_679_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( 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_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_680_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_681_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_682_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_683_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_684_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_685_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_686_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_687_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_688_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_689_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_690_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_691_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_692_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_693_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_694_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_695_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_696_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_697_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( 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_num,aa_num_num(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_698_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_699_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_700_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_701_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_702_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_703_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_704_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_705_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_706_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_707_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_708_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_709_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_710_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_711_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_712_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_713_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_714_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_715_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_716_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_717_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_718_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_719_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_720_linorder__le__less__linear,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,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_721_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( 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,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_722_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_723_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_724_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_725_order__le__imp__less__or__eq,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 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_726_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_727_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( 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,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_728_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_729_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_730_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_731_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_732_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_733_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_734_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_735_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_736_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_737_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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_738_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,aa_num_num(bit0,one))),N))) ).

% of_nat_less_two_power
tff(fact_739_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_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_740_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_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_741_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)),aa_num_real(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_742_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_743_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_744_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( some_nat(X2) = some_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_745_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_746_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_747_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_748_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_749_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_750_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_751_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_752_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_753_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_754_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% real_arch_simple
tff(fact_755_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X5)) ) ).

% minf(8)
tff(fact_756_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X5)) ) ).

% minf(8)
tff(fact_757_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X5)) ) ).

% minf(8)
tff(fact_758_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X5)) ) ).

% minf(8)
tff(fact_759_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5)) ) ).

% minf(8)
tff(fact_760_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_761_int__if,axiom,
    ! [P: bool,A: nat,B: nat] :
      ( ( pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,A) ) )
      & ( ~ pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ) ).

% int_if
tff(fact_762_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X5: real] : pp(aa_set_real_bool(member_real(X5),S2))
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z3)) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Y3)) )
            & ! [Z3: real] :
                ( ! [X4: real] :
                    ( pp(aa_set_real_bool(member_real(X4),S2))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z3)) ) ) ) ) ).

% complete_real
tff(fact_763_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
      | ( A = X )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A)) ) ).

% verit_la_generic
tff(fact_764_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_765_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_766_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
           => ( ( pp(aa_real_bool(P,X5))
                & pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                & pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_767_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
           => ( ( pp(aa_rat_bool(P,X5))
                & pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                & pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_768_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
           => ( ( pp(aa_num_bool(P,X5))
                & pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                & pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_769_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
           => ( ( pp(aa_nat_bool(P,X5))
                & pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                & pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_770_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
           => ( ( pp(aa_int_bool(P,X5))
                & pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                & pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_771_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
           => ( ( pp(aa_real_bool(P,X5))
                | pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                | pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_772_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
           => ( ( pp(aa_rat_bool(P,X5))
                | pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                | pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_773_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
           => ( ( pp(aa_num_bool(P,X5))
                | pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                | pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_774_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
           => ( ( pp(aa_nat_bool(P,X5))
                | pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                | pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_775_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
           => ( ( pp(aa_int_bool(P,X5))
                | pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                | pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_776_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_777_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_778_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_779_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_780_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_781_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_782_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_783_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_784_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_785_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_786_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),T)) ) ).

% pinf(5)
tff(fact_787_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),T)) ) ).

% pinf(5)
tff(fact_788_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),T)) ) ).

% pinf(5)
tff(fact_789_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),T)) ) ).

% pinf(5)
tff(fact_790_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T)) ) ).

% pinf(5)
tff(fact_791_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X5)) ) ).

% pinf(7)
tff(fact_792_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X5)) ) ).

% pinf(7)
tff(fact_793_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X5)) ) ).

% pinf(7)
tff(fact_794_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X5)) ) ).

% pinf(7)
tff(fact_795_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5)) ) ).

% pinf(7)
tff(fact_796_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
           => ( ( pp(aa_real_bool(P,X5))
                & pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                & pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_797_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
           => ( ( pp(aa_rat_bool(P,X5))
                & pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                & pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_798_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
           => ( ( pp(aa_num_bool(P,X5))
                & pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                & pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_799_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
           => ( ( pp(aa_nat_bool(P,X5))
                & pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                & pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_800_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
           => ( ( pp(aa_int_bool(P,X5))
                & pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                & pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_801_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q2,X4)) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
           => ( ( pp(aa_real_bool(P,X5))
                | pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                | pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_802_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q2,X4)) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
           => ( ( pp(aa_rat_bool(P,X5))
                | pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                | pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_803_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q2,X4)) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
           => ( ( pp(aa_num_bool(P,X5))
                | pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                | pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_804_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q2,X4)) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
           => ( ( pp(aa_nat_bool(P,X5))
                | pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                | pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_805_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q2,X4)) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
           => ( ( pp(aa_int_bool(P,X5))
                | pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                | pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_806_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_807_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_808_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_809_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_810_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_811_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_812_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_813_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_814_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_815_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_816_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),T)) ) ).

% minf(5)
tff(fact_817_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),T)) ) ).

% minf(5)
tff(fact_818_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),T)) ) ).

% minf(5)
tff(fact_819_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),T)) ) ).

% minf(5)
tff(fact_820_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T)) ) ).

% minf(5)
tff(fact_821_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X5)) ) ).

% minf(7)
tff(fact_822_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X5)) ) ).

% minf(7)
tff(fact_823_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X5)) ) ).

% minf(7)
tff(fact_824_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X5)) ) ).

% minf(7)
tff(fact_825_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5)) ) ).

% minf(7)
tff(fact_826_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_827_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),T)) ) ).

% pinf(6)
tff(fact_828_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),T)) ) ).

% pinf(6)
tff(fact_829_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),T)) ) ).

% pinf(6)
tff(fact_830_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),T)) ) ).

% pinf(6)
tff(fact_831_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T)) ) ).

% pinf(6)
tff(fact_832_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X5)) ) ).

% pinf(8)
tff(fact_833_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X5)) ) ).

% pinf(8)
tff(fact_834_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X5)) ) ).

% pinf(8)
tff(fact_835_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X5)) ) ).

% pinf(8)
tff(fact_836_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5)) ) ).

% pinf(8)
tff(fact_837_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),T)) ) ).

% minf(6)
tff(fact_838_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),T)) ) ).

% minf(6)
tff(fact_839_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),T)) ) ).

% minf(6)
tff(fact_840_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),T)) ) ).

% minf(6)
tff(fact_841_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T)) ) ).

% minf(6)
tff(fact_842_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B))
         => ? [C2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
              & ! [X5: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X5))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),C2)) )
                 => pp(aa_real_bool(P,X5)) )
              & ! [D: real] :
                  ( ! [X4: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),D)) )
                     => pp(aa_real_bool(P,X4)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_843_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B))
         => ? [C2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
              & ! [X5: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X5))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),C2)) )
                 => pp(aa_nat_bool(P,X5)) )
              & ! [D: nat] :
                  ( ! [X4: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D)) )
                     => pp(aa_nat_bool(P,X4)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_844_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B))
         => ? [C2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
              & ! [X5: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X5))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),C2)) )
                 => pp(aa_int_bool(P,X5)) )
              & ! [D: int] :
                  ( ! [X4: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D)) )
                     => pp(aa_int_bool(P,X4)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C2)) ) ) ) ) ) ).

% complete_interval
tff(fact_845_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,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_846_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,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_847_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,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_848_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,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_849_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,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_850_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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_851_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,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_852_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,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_853_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,aa_num_num(bit0,one)))),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_854_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,aa_num_num(bit0,one)))),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_855_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_856_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,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_857_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_858_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,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_859_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_860_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_861_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_862_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_863_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,aa_num_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_864_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(aa_num_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_865_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_866_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_num_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_867_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(aa_num_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_868_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_869_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_870_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_871_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_872_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_889_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_890_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_891_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_892_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_893_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_894_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_895_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_896_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_925_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_926_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_927_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_928_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_929_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,Z)) = aa_int_complex(ring_17405671764205052669omplex,Z) ) ) ).

% of_nat_nat
tff(fact_978_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_979_of__nat__nat,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z)) = aa_int_rat(ring_1_of_int_rat,Z) ) ) ).

% of_nat_nat
tff(fact_980_of__nat__nat,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)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_981_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_982_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_983_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_984_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_985_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_986_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_987_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_988_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_989_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_990_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_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_991_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_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_992_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_993_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)),aa_num_real(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_994_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)),aa_num_rat(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_995_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_996_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)),aa_num_real(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_997_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)),aa_num_rat(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_998_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_999_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_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_1000_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_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_1001_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_1002_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_1003_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_num_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_1004_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_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_1005_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_1006_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_1007_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,aa_num_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_1008_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(aa_num_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_num_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_1025_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(aa_num_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_1026_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_1027_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,aa_num_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_1028_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(aa_num_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_1029_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_1030_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_1031_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: 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),Z4))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_1032_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% all_nat
tff(fact_1033_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% ex_nat
tff(fact_1034_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N3: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => pp(aa_nat_bool(P,N3)) )
        & ( 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_1044_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_1045_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_1046_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_1047_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_1048_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_le_of_int
tff(fact_1049_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_le_of_int
tff(fact_1050_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2))) ).

% ex_less_of_int
tff(fact_1051_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% ex_less_of_int
tff(fact_1052_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X)) ).

% ex_of_int_less
tff(fact_1053_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)) ).

% ex_of_int_less
tff(fact_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1068_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))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1069_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_1070_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_1071_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_1072_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_1073_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_1074_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_1075_power__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 ) ) ).

% power_not_zero
tff(fact_1076_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_1077_power__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 ) ) ).

% power_not_zero
tff(fact_1078_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_1095_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_1096_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_1097_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_1098_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_1099_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N)) ).

% i0_lb
tff(fact_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_nat__numeral__as__int,axiom,
    ! [X5: num] : aa_num_nat(numeral_numeral_nat,X5) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X5)) ).

% nat_numeral_as_int
tff(fact_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_1118_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_1119_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_1120_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_1121_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_le_numeral
tff(fact_1122_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_le_numeral
tff(fact_1123_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_1124_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_1125_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_less_numeral
tff(fact_1126_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_less_numeral
tff(fact_1127_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_1128_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_1129_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_1130_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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))
            & ! [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2))
               => ~ pp(aa_nat_bool(P,I2)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_1154_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_1155_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_1156_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_1157_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_1158_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_1159_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A)) ) ).

% ex_gt_or_lt
tff(fact_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_1168_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_1169_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_1170_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_1171_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_1172_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,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_1173_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,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_1174_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,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_1175_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,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1176_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1177_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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1178_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1179_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1180_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,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1181_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1182_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,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_1183_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,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_1184_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,aa_num_num(bit0,one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_1185_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,aa_num_num(bit0,one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_1186_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,aa_num_num(bit0,one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_1187_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_1188_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_1189_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_1190_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))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),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 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1191_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_1192_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_1193_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_1194_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_1195_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_1196_zero__reorient,axiom,
    ! [X: literal] :
      ( ( zero_zero_literal = X )
    <=> ( X = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_1197_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1198_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1199_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1200_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1201_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_1202_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,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_1203_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,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_1204_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,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_1205_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_1206_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,aa_num_num(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,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1207_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,aa_num_num(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,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1208_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,aa_num_num(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,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1209_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_1210_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,aa_num_num(bit0,one))) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_1211_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,aa_num_num(bit0,one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_1212_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,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_1213_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_1214_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_1215_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_1216_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_1217_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,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_1218_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_1219_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_1220_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_1221_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_1222_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_1223_or_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),A) = A ).

% or.idem
tff(fact_1224_or_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),A) = A ).

% or.idem
tff(fact_1225_or_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se1409905431419307370or_int(A),aa_int_int(bit_se1409905431419307370or_int(A),B)) = aa_int_int(bit_se1409905431419307370or_int(A),B) ).

% or.left_idem
tff(fact_1226_or_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ).

% or.left_idem
tff(fact_1227_or_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se1409905431419307370or_int(A),B)),B) = aa_int_int(bit_se1409905431419307370or_int(A),B) ).

% or.right_idem
tff(fact_1228_or_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ).

% or.right_idem
tff(fact_1229_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_1230_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_1231_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_1232_bit_Oxor__left__self,axiom,
    ! [X: int,Y: int] : aa_int_int(bit_se6526347334894502574or_int(X),aa_int_int(bit_se6526347334894502574or_int(X),Y)) = Y ).

% bit.xor_left_self
tff(fact_1233_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_1234_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_1235_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_1236_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_1237_power__one,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(one_one_complex),N) = one_one_complex ).

% power_one
tff(fact_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_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_1248_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_1249_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_1250_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_1251_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_1252_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_1253_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_1254_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_1255_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_1256_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_1257_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_1258_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_1259_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_1260_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_1261_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_1262_take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_1263_take__bit__of__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_1264_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_1265_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_1266_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_1267_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_1268_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_1269_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_1270_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_1271_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_1272_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_1273_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_1274_bit_Oxor__self,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),X) = zero_zero_int ).

% bit.xor_self
tff(fact_1275_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_1284_take__bit__or,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se1409905431419307370or_int(A),B)) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(bit_se2923211474154528505it_int(N),B)) ).

% take_bit_or
tff(fact_1285_take__bit__or,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B)) ).

% take_bit_or
tff(fact_1286_take__bit__xor,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se6526347334894502574or_int(A),B)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(bit_se2923211474154528505it_int(N),B)) ).

% take_bit_xor
tff(fact_1287_take__bit__xor,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B)) ).

% take_bit_xor
tff(fact_1288_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1289_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_real(numeral_numeral_real,N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1290_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_rat(numeral_numeral_rat,N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_1291_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_1292_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_1293_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1294_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1295_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_1303_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_1304_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_1305_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_1306_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_1307_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_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_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_1322_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_1323_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_1324_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_1325_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_1326_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_1327_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_1328_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_1329_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_1330_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_1331_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_1332_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_1333_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_1334_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_1335_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_1336_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_1337_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_1338_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_1339_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_1340_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_1341_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_1342_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_1343_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_1344_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_1345_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_real(ring_1_of_int_real,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_1346_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_rat(ring_1_of_int_rat,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_1347_of__nat__nat__take__bit__eq,axiom,
    ! [N: nat,K: int] : aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_1348_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_1349_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_1350_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_1351_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_1352_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_1353_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_1354_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_1355_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_1356_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_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_1357_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_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_1358_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_1359_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_1360_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_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_1361_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_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_1362_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_1363_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_1364_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_1365_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(N,A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_1366_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_1367_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)))
    <=> ( ( N = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ) ).

% even_take_bit_eq
tff(fact_1368_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_1369_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_1370_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_1371_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,aa_num_num(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,aa_num_num(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_1372_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,aa_num_num(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,aa_num_num(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_1373_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,aa_num_num(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,aa_num_num(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_1374_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,aa_num_num(bit0,one))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_1375_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,aa_num_num(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_1376_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,aa_num_num(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_1377_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,aa_num_num(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_1378_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,aa_num_num(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_1379_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,aa_num_num(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_1380_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_1381_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_1382_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),N)) ) ).

% even_of_nat
tff(fact_1383_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,aa_num_num(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,aa_num_num(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_1384_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,aa_num_num(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,aa_num_num(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_1385_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,aa_num_num(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,aa_num_num(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_1386_or_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se1409905431419307370or_int(A),B)),C) = aa_int_int(bit_se1409905431419307370or_int(A),aa_int_int(bit_se1409905431419307370or_int(B),C)) ).

% or.assoc
tff(fact_1387_or_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.assoc
tff(fact_1388_xor_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se6526347334894502574or_int(A),B)),C) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se6526347334894502574or_int(B),C)) ).

% xor.assoc
tff(fact_1389_xor_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),C)) ).

% xor.assoc
tff(fact_1390_or_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se1409905431419307370or_int(A),B) = aa_int_int(bit_se1409905431419307370or_int(B),A) ).

% or.commute
tff(fact_1391_or_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),A) ).

% or.commute
tff(fact_1392_xor_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se6526347334894502574or_int(A),B) = aa_int_int(bit_se6526347334894502574or_int(B),A) ).

% xor.commute
tff(fact_1393_xor_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),A) ).

% xor.commute
tff(fact_1394_or_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se1409905431419307370or_int(B),aa_int_int(bit_se1409905431419307370or_int(A),C)) = aa_int_int(bit_se1409905431419307370or_int(A),aa_int_int(bit_se1409905431419307370or_int(B),C)) ).

% or.left_commute
tff(fact_1395_or_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.left_commute
tff(fact_1396_xor_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se6526347334894502574or_int(B),aa_int_int(bit_se6526347334894502574or_int(A),C)) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se6526347334894502574or_int(B),C)) ).

% xor.left_commute
tff(fact_1397_xor_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),C)) ).

% xor.left_commute
tff(fact_1398_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_1399_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_1400_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_1401_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_1402_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_1403_of__nat__or__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N)) = aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_or_eq
tff(fact_1404_of__nat__or__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_or_eq
tff(fact_1405_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).

% take_bit_of_nat
tff(fact_1406_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).

% take_bit_of_nat
tff(fact_1407_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_1408_take__bit__of__int,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_of_int
tff(fact_1409_of__int__or__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se1409905431419307370or_int(K),L)) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_or_eq
tff(fact_1410_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N)) = aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_xor_eq
tff(fact_1411_of__nat__xor__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_xor_eq
tff(fact_1412_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_1413_of__int__xor__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se6526347334894502574or_int(K),L)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_xor_eq
tff(fact_1414_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_1415_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_1416_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_1417_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_1418_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_1419_bit_Odisj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se1409905431419307370or_int(X),zero_zero_int) = X ).

% bit.disj_zero_right
tff(fact_1420_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se1409905431419307370or_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_1421_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_1422_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(bit_se2923211474154528505it_int(N),B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_int_int(bit_se2923211474154528505it_int(M),A) = aa_int_int(bit_se2923211474154528505it_int(M),B) ) ) ) ).

% take_bit_tightened
tff(fact_1423_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_1424_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_1425_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_1426_odd__one,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer)) ).

% odd_one
tff(fact_1427_odd__one,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat)) ).

% odd_one
tff(fact_1428_odd__one,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int)) ).

% odd_one
tff(fact_1429_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_1430_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_1431_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_1432_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_1433_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_1434_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_1435_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_1436_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( pp(aa_complex_bool(dvd_dvd_complex(X),Y))
     => pp(aa_complex_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_1437_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_1438_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_1439_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_1440_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_1441_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_1442_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_1443_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_1444_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_1445_even__or__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1080825931792720795nteger(A,B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% even_or_iff
tff(fact_1446_even__or__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_or_iff
tff(fact_1447_even__or__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_or_iff
tff(fact_1448_even__xor__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3222712562003087583nteger(A,B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% even_xor_iff
tff(fact_1449_even__xor__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_xor_iff
tff(fact_1450_even__xor__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_xor_iff
tff(fact_1451_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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(dvd_dvd_complex(X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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_1467_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_1468_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_1469_even__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_1470_even__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_1471_even__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) ).

% even_numeral
tff(fact_1472_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_1473_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_1474_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_1475_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_1476_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_1477_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_1478_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_1479_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_1480_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_1481_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_1482_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(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_1483_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_1484_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_1485_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_1486_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_1487_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: nat] :
      ( pp(aa_complex_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(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_1488_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_1489_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_1490_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_1491_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_1492_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( pp(aa_complex_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(dvd_dvd_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).

% dvd_power_le
tff(fact_1493_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_1494_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_1495_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_1496_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_1497_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_1498_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N))) ).

% one_le_numeral
tff(fact_1499_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% one_le_numeral
tff(fact_1500_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_1501_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_1502_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_1503_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_1504_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_1505_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_1506_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_1507_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_1508_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_1509_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_1510_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_1511_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_1512_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_1513_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_1514_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_1515_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_1516_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_1517_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),A)) ) ).

% take_bit_eq_0_iff
tff(fact_1518_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_1519_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_1520_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_1521_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_1522_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_1523_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_1524_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_1525_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_1526_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_1527_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,aa_num_num(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,aa_num_num(bit0,one))),K)) ) ).

% even_of_int_iff
tff(fact_1528_even__zero,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger)) ).

% even_zero
tff(fact_1529_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat)) ).

% even_zero
tff(fact_1530_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int)) ).

% even_zero
tff(fact_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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,aa_num_num(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_1556_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,aa_num_num(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_1557_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,aa_num_num(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_1558_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_1559_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_1560_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,aa_num_num(bit0,one))),N) )
     => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_1561_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_1562_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_1563_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_1564_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_1565_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_1566_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_1567_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_1568_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_1569_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_1570_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_1571_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_1572_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_1573_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_1574_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_1575_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_1576_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_1577_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_1578_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_1579_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_1580_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_1581_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_1582_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_1583_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_1584_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_1585_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_1586_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,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_1587_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,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_1588_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,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_1589_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,aa_num_num(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_1590_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,aa_num_num(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_1591_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,aa_num_num(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_1592_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,aa_num_num(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_1593_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,aa_num_num(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_1594_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,aa_num_num(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_1595_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,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_1596_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,aa_num_num(bit0,one))))) ) ).

% sqrt_le_D
tff(fact_1597_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,aa_num_num(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_1598_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,aa_num_num(bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_1599_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,aa_num_num(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,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_1600_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,aa_num_num(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,aa_num_num(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_1601_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,aa_num_num(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,aa_num_num(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_1602_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,aa_num_num(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_1603_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_1604_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,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_1605_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,aa_num_num(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_1606_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,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_1607_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,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_1608_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,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_1609_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,aa_num_num(bit0,one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_1610_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,aa_num_num(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,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1611_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,aa_num_num(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,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1612_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,aa_num_num(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,aa_num_num(bit0,one))),N))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1613_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_1614_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_1615_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_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_1626_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_1627_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_1628_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_1629_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_1630_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_1631_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ) ).

% log_ceil_idem
tff(fact_1632_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1633_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1634_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1635_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1636_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_1637_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_1638_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_1639_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_1640_ceiling__of__int,axiom,
    ! [Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_int_rat(ring_1_of_int_rat,Z)) = Z ).

% ceiling_of_int
tff(fact_1641_ceiling__of__int,axiom,
    ! [Z: int] : aa_real_int(archim7802044766580827645g_real,aa_int_real(ring_1_of_int_real,Z)) = Z ).

% ceiling_of_int
tff(fact_1642_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)) = X )
    <=> ? [N3: int] : X = aa_int_rat(ring_1_of_int_rat,N3) ) ).

% of_int_ceiling_cancel
tff(fact_1643_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)) = X )
    <=> ? [N3: int] : X = aa_int_real(ring_1_of_int_real,N3) ) ).

% of_int_ceiling_cancel
tff(fact_1644_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_1645_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_1646_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_1647_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_1648_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_1649_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_1650_ceiling__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_1651_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_1652_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_1653_ceiling__of__nat,axiom,
    ! [N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% ceiling_of_nat
tff(fact_1654_ceiling__of__nat,axiom,
    ! [N: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% ceiling_of_nat
tff(fact_1655_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_1656_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_1657_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_1658_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_1659_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,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_1660_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_1661_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_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_1662_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_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_1663_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_1664_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),aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_numeral
tff(fact_1665_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_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),aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_numeral
tff(fact_1666_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_1667_ceiling__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_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_1668_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,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_1669_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)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_less_ceiling
tff(fact_1670_one__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_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_1671_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_1672_ceiling__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% ceiling_le_one
tff(fact_1673_ceiling__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).

% ceiling_le_one
tff(fact_1674_one__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_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_1675_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_1676_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_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_1677_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_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_1678_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_1679_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_1680_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: 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),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q3))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_1681_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_mono
tff(fact_1682_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_mono
tff(fact_1683_le__of__int__ceiling,axiom,
    ! [X: real] : 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)))) ).

% le_of_int_ceiling
tff(fact_1684_le__of__int__ceiling,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)))) ).

% le_of_int_ceiling
tff(fact_1685_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_1686_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_1687_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_1688_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_1689_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_1690_of__nat__ceiling,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2))))) ).

% of_nat_ceiling
tff(fact_1691_of__nat__ceiling,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R2))))) ).

% of_nat_ceiling
tff(fact_1692_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_1693_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,aa_rat_int(archim2889992004027027881ng_rat,X)),A)) ) ).

% ceiling_le
tff(fact_1694_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% ceiling_le_iff
tff(fact_1695_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% ceiling_le_iff
tff(fact_1696_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(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_1697_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_1698_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_1699_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_1700_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_1701_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_1702_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_1703_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_1704_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_1705_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_1706_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,aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_1707_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,aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_1708_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,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_1709_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,aa_num_num(bit0,one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_1710_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_1711_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1712_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1713_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1714_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1715_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_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,aa_num_num(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,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_1727_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_1728_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_1729_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_1730_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_1731_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_1732_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_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_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_1742_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_1743_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_1744_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_1745_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_1746_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_1747_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1748_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1749_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1750_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A)))
    <=> ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1751_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1752_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1753_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1754_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1755_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1756_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(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_1757_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_1758_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,aa_num_num(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(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_1759_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1760_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,aa_num_num(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,aa_num_num(bit0,one))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1761_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1762_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,aa_num_num(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,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_1763_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,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_1764_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,aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_le_ceiling
tff(fact_1765_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)),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_le_ceiling
tff(fact_1766_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_1767_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1768_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_1769_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_1770_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_1771_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1772_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_1773_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_1774_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_1775_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_1776_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_1777_idiff__0__right,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_1778_idiff__0,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat),N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_1779_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,aa_num_num(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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_1780_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_1781_add__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(plus_plus_rat(A),C)),aa_rat_rat(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_1782_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_1783_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(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_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_1784_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_1785_add__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(plus_plus_rat(C),A)),aa_rat_rat(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_1786_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_1787_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(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_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_1788_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_1789_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1790_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1791_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_1792_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_1793_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_1794_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_1795_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_1796_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_1797_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_1798_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_1799_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1800_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_1801_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1802_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_1803_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1804_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_1805_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1806_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_1807_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1808_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_1809_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1810_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_1811_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1812_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_1813_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1814_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_1815_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1816_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1817_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_1818_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_1819_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1820_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_1821_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1822_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_1823_add__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(plus_plus_rat(C),A)),aa_rat_rat(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_1824_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_1825_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(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_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_1826_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_1827_add__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(plus_plus_rat(A),C)),aa_rat_rat(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_1828_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_1829_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(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_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_1830_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_1831_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1832_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_1833_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1834_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_1835_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_1836_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_1837_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_1838_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_1839_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_1840_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_1841_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_1842_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_1843_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_1844_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_1845_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_1846_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1847_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_1848_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1849_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1850_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_1851_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1852_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_1853_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1854_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1855_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_1856_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_1857_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_1858_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1859_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_1860_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_1861_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_1862_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_1863_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_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_1864_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_1865_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1866_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1867_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_1868_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_1869_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_1870_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_1871_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_1872_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_1873_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_1874_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_1875_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_1876_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_1877_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_1878_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_1879_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_1880_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1881_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_1882_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1883_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_1884_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_1885_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_1886_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_1887_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_1888_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1889_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_1890_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1891_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_1892_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% bits_div_by_1
tff(fact_1893_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_1894_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_1895_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)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_1896_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_1897_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_1898_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_1899_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_1900_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_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_1911_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_1912_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(aa_int_fun_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_1913_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_1914_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_1915_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_1916_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(aa_int_fun_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_1917_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_1918_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(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_1919_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_1920_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(aa_int_fun_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_1921_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_1922_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(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_1923_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_1924_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(aa_int_fun_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_1925_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_1926_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,aa_rat_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_1927_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(aa_int_fun_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_1928_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_1929_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),aa_rat_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_1930_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(aa_int_fun_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_1931_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_1932_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_1933_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_1934_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_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_1935_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_1936_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_1937_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_1938_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_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_1939_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_1940_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(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_1941_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_1942_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(aa_int_fun_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_1943_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_1944_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(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_1945_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_1946_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(aa_int_fun_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_1947_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_1948_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_1949_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(aa_int_fun_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_1950_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_1951_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),aa_rat_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_1952_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(aa_int_fun_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_1953_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_1954_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),aa_rat_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_1955_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(aa_int_fun_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_1956_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_1957_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),aa_rat_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_1958_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(aa_int_fun_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_1959_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_1960_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_1961_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1962_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_1963_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_1964_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1965_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_1966_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(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1967_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_1968_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1969_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_1970_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1971_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_1972_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_1973_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_1974_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_1975_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_1976_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_1977_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_1978_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_1979_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_1980_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_1981_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_1982_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_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_1983_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_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_1984_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_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_1985_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(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_1986_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_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_1987_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_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_1988_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_rat_rat(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_1989_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_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_1990_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_1991_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1992_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1993_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_1994_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_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_1995_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_1996_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1997_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1998_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_1999_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_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_2000_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_2001_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_2002_diff__nat__numeral,axiom,
    ! [V: num,V2: 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,V2)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V2))) ).

% diff_nat_numeral
tff(fact_2003_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(aa_int_fun_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_2004_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z) ).

% ceiling_add_of_int
tff(fact_2005_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(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),Z) ).

% ceiling_add_of_int
tff(fact_2006_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(aa_int_fun_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_2007_ceiling__diff__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z) ).

% ceiling_diff_of_int
tff(fact_2008_ceiling__diff__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),Z) ).

% ceiling_diff_of_int
tff(fact_2009_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2010_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2011_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2012_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,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2013_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2014_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
    <=> ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_2015_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_2016_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_2017_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_2018_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,aa_num_num(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,aa_num_num(bit0,one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_2019_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_2020_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_2021_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_2022_ceiling__add__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_add_one
tff(fact_2023_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(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_2024_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_2025_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_2026_ceiling__diff__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_diff_one
tff(fact_2027_ceiling__diff__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_diff_one
tff(fact_2028_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,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2029_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2030_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,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2031_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2032_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2033_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2034_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2035_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer)))
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_2036_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,aa_num_num(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,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_2037_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_2038_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ).

% even_diff
tff(fact_2039_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ).

% even_diff
tff(fact_2040_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(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_2041_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_2042_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,aa_num_num(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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_2043_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,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_2044_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_2045_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_2046_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,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_2047_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_2048_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,aa_num_num(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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_2049_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,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_2050_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,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_2051_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_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),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_2052_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(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(aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_2053_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_2054_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,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_2055_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_2056_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))
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2057_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_2058_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(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2059_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: 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,J),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,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2060_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(plus_plus_rat(J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2061_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: 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,J),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,J),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)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2062_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2063_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_2064_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,aa_rat_rat(plus_plus_rat(I),K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_2065_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_2066_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(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_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_2076_diff__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 ) ) ).

% diff_add
tff(fact_2077_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_2078_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C)) ) ).

% le_diff_eq
tff(fact_2079_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(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% le_diff_eq
tff(fact_2080_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_2081_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% diff_le_eq
tff(fact_2082_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_le_eq
tff(fact_2083_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_2084_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% diff_less_eq
tff(fact_2085_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(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_less_eq
tff(fact_2086_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_2087_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C)) ) ).

% less_diff_eq
tff(fact_2088_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(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% less_diff_eq
tff(fact_2089_ab__semigroup__add__class_Oadd__ac_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)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2090_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2091_ab__semigroup__add__class_Oadd__ac_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,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)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2092_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_2093_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2094_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2095_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( 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,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2096_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2097_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A3),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_2098_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A3),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_2099_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),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_2100_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2101_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B4) = 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_2102_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B4) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_2103_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B: nat,A: nat] :
      ( ( B4 = 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),B4) = 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_2104_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2105_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A3),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_2106_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A3),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2107_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2108_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_2109_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_2110_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_2111_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_2112_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_2113_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_2114_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_2115_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_2116_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_2117_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_2118_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_2119_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_2120_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_2121_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_2122_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_2123_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_2124_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_2125_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_2126_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_2127_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_2128_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2129_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2130_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D3) )
     => ( ( A = B )
      <=> ( C = D3 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2131_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D3: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D3) )
     => ( ( A = B )
      <=> ( C = D3 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2132_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D3) )
     => ( ( A = B )
      <=> ( C = D3 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2133_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_2134_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2135_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2136_add_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) ).

% add.commute
tff(fact_2137_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_2138_add_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) ).

% add.commute
tff(fact_2139_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% add.commute
tff(fact_2140_add_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)) ).

% add.left_commute
tff(fact_2141_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_2142_add_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)) ).

% add.left_commute
tff(fact_2143_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_2144_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_2145_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2146_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_2147_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2148_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_2149_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2150_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2151_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_2152_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2153_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_2154_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2155_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_2156_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_2157_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_2158_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_2159_diff__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) ).

% diff_right_commute
tff(fact_2160_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_2161_diff__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) ).

% diff_right_commute
tff(fact_2162_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_2163_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_2164_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_2165_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_2166_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_2167_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D3: 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),D3)) = 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),D3)) ).

% add_diff_add
tff(fact_2168_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D3: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D3)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D3)) ).

% add_diff_add
tff(fact_2169_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D3)) ).

% add_diff_add
tff(fact_2170_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_2171_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_2172_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_2173_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_2174_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(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_2175_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_2176_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2177_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_2178_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2179_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_2180_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_2181_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)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_2182_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_2183_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(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_2184_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(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_2185_int__minus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_2186_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y)))) ).

% ceiling_add_le
tff(fact_2187_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(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).

% ceiling_add_le
tff(fact_2188_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D3) )
     => ( 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),D3)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2189_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D3: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D3) )
     => ( 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),D3)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2190_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D3) )
     => ( 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),D3)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2191_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_2192_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,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_right_mono
tff(fact_2193_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(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_right_mono
tff(fact_2194_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_2195_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,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_left_mono
tff(fact_2196_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(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_left_mono
tff(fact_2197_diff__mono,axiom,
    ! [A: real,B: real,D3: 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,D3),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),D3))) ) ) ).

% diff_mono
tff(fact_2198_diff__mono,axiom,
    ! [A: rat,B: rat,D3: 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,D3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D3))) ) ) ).

% diff_mono
tff(fact_2199_diff__mono,axiom,
    ! [A: int,B: int,D3: 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,D3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D3))) ) ) ).

% diff_mono
tff(fact_2200_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_2201_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2202_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_2203_diff__strict__mono,axiom,
    ! [A: real,B: real,D3: 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,D3),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),D3))) ) ) ).

% diff_strict_mono
tff(fact_2204_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D3: 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,D3),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D3))) ) ) ).

% diff_strict_mono
tff(fact_2205_diff__strict__mono,axiom,
    ! [A: int,B: int,D3: 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,D3),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D3))) ) ) ).

% diff_strict_mono
tff(fact_2206_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D3) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D3)) ) ) ).

% diff_eq_diff_less
tff(fact_2207_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D3: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D3) )
     => ( 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),D3)) ) ) ).

% diff_eq_diff_less
tff(fact_2208_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D3) )
     => ( 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),D3)) ) ) ).

% diff_eq_diff_less
tff(fact_2209_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_2210_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,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_strict_left_mono
tff(fact_2211_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(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_strict_left_mono
tff(fact_2212_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_2213_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,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_strict_right_mono
tff(fact_2214_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(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_strict_right_mono
tff(fact_2215_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_2216_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,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(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_2217_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_2218_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(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_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_2219_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_2220_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,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(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_2221_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_2222_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(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_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_2223_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_2224_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_2225_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,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))) ) ).

% add_right_mono
tff(fact_2226_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_2227_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(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_right_mono
tff(fact_2228_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_2229_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_2230_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,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% add_left_mono
tff(fact_2231_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_2232_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(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_left_mono
tff(fact_2233_add__mono,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_mono
tff(fact_2234_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D3))) ) ) ).

% add_mono
tff(fact_2235_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_mono
tff(fact_2236_add__mono,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D3))) ) ) ).

% add_mono
tff(fact_2237_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2238_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & 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,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2239_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2240_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & 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(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2241_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & 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,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2242_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & 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,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2243_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & 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,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2244_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & 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(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2245_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( 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,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2246_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2247_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( 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,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2248_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2249_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_2250_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2251_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_2252_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_2253_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_2254_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_2255_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_2256_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_2257_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2258_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_2259_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_2260_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_2261_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2262_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_2263_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2264_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2265_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & 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,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2266_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2267_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & 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(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2268_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2269_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & 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,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2270_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2271_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & 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(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2272_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( 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,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2273_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2274_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( 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,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2275_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2276_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D3))
       => 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),D3))) ) ) ).

% add_strict_mono
tff(fact_2277_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D3))) ) ) ).

% add_strict_mono
tff(fact_2278_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_strict_mono
tff(fact_2279_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D3))) ) ) ).

% add_strict_mono
tff(fact_2280_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_2281_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,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% add_strict_left_mono
tff(fact_2282_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_2283_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(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_strict_left_mono
tff(fact_2284_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_2285_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,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))) ) ).

% add_strict_right_mono
tff(fact_2286_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_2287_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(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_strict_right_mono
tff(fact_2288_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_2289_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(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_2290_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_2291_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(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_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_2292_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_2293_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(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_2294_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_2295_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(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_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_2296_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_2297_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_2298_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B)),N) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).

% power_divide
tff(fact_2299_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_2300_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_2301_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_2302_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_2303_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_2304_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_2305_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_2306_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_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_2314_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_2315_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M4)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_2316_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_2317_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(bit_se2923211474154528505it_int(N),B))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_2318_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_2319_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(aa_int_fun_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_2320_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_2321_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_2322_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_2323_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_2324_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_2325_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_2326_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_2327_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_2328_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)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_2329_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_2330_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(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_2331_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(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).

% int_ops(6)
tff(fact_2332_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z4),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_2333_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(aa_int_fun_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_2334_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_2335_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_2336_dbl__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_numeral_dbl_rat,X) = aa_rat_rat(plus_plus_rat(X),X) ).

% dbl_def
tff(fact_2337_dbl__def,axiom,
    ! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X) ).

% dbl_def
tff(fact_2338_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_2339_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_2340_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_2341_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_2342_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),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_2343_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_2344_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(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(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_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_2345_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_2346_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(aa_int_fun_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_2347_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).

% even_diff_iff
tff(fact_2348_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_2349_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,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_2350_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(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_2351_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_2352_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,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_2353_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(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_2354_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_2355_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_2356_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_2357_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_2358_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,aa_rat_rat(plus_plus_rat(A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2359_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_2360_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(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2361_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_2362_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),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_increasing
tff(fact_2363_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_2364_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(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing
tff(fact_2365_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_2366_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,aa_rat_rat(plus_plus_rat(A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2367_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_2368_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(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2369_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_2370_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),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_increasing2
tff(fact_2371_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_2372_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(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing2
tff(fact_2373_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_2374_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),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2375_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_2376_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(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2377_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_2378_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,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_2379_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_2380_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(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_2381_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_2382_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))
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2383_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_2384_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(aa_int_fun_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_2385_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_2386_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))
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2387_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_2388_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(aa_int_fun_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_2389_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_less_le_mono
tff(fact_2390_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D3))) ) ) ).

% add_less_le_mono
tff(fact_2391_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_less_le_mono
tff(fact_2392_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D3))) ) ) ).

% add_less_le_mono
tff(fact_2393_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_le_less_mono
tff(fact_2394_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D3))) ) ) ).

% add_le_less_mono
tff(fact_2395_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => 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),D3))) ) ) ).

% add_le_less_mono
tff(fact_2396_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D3))) ) ) ).

% add_le_less_mono
tff(fact_2397_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2398_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & 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,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2399_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2400_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & 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(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2401_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2402_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & 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,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2403_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & 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,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2404_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & 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(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2405_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_2406_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,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_2407_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_2408_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(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_2409_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_2410_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),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_pos_pos
tff(fact_2411_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_2412_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(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_pos
tff(fact_2413_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_2414_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_2415_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),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% pos_add_strict
tff(fact_2416_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_2417_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(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% pos_add_strict
tff(fact_2418_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_2419_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_2420_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_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_2421_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_2422_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_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_2423_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_2424_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2425_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_2426_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(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2427_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_2428_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_2429_power__one__over,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),N) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),N)) ).

% power_one_over
tff(fact_2430_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_2431_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,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat))) ) ).

% add_mono1
tff(fact_2432_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_2433_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(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int))) ) ).

% add_mono1
tff(fact_2434_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_2435_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat))) ).

% less_add_one
tff(fact_2436_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_2437_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_2438_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_2439_numeral__Bit0,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_Bit0
tff(fact_2440_numeral__Bit0,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_Bit0
tff(fact_2441_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(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_2442_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_2443_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_2444_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2445_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_2446_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_2447_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2448_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_2449_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_2450_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(divide_divide_int(A),B)),N) = aa_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_2451_pinf_I9_J,axiom,
    ! [D3: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X5))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S))) ) ) ).

% pinf(9)
tff(fact_2452_pinf_I9_J,axiom,
    ! [D3: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S))) ) ) ).

% pinf(9)
tff(fact_2453_pinf_I9_J,axiom,
    ! [D3: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S))) ) ) ).

% pinf(9)
tff(fact_2454_pinf_I9_J,axiom,
    ! [D3: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S))) ) ) ).

% pinf(9)
tff(fact_2455_pinf_I9_J,axiom,
    ! [D3: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S))) ) ) ).

% pinf(9)
tff(fact_2456_pinf_I10_J,axiom,
    ! [D3: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X5))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S))) ) ) ).

% pinf(10)
tff(fact_2457_pinf_I10_J,axiom,
    ! [D3: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X5))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S))) ) ) ).

% pinf(10)
tff(fact_2458_pinf_I10_J,axiom,
    ! [D3: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X5))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S))) ) ) ).

% pinf(10)
tff(fact_2459_pinf_I10_J,axiom,
    ! [D3: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X5))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S))) ) ) ).

% pinf(10)
tff(fact_2460_pinf_I10_J,axiom,
    ! [D3: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X5))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S))) ) ) ).

% pinf(10)
tff(fact_2461_minf_I9_J,axiom,
    ! [D3: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),Z2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S))) ) ) ).

% minf(9)
tff(fact_2462_minf_I9_J,axiom,
    ! [D3: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S))) ) ) ).

% minf(9)
tff(fact_2463_minf_I9_J,axiom,
    ! [D3: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S))) ) ) ).

% minf(9)
tff(fact_2464_minf_I9_J,axiom,
    ! [D3: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S))) ) ) ).

% minf(9)
tff(fact_2465_minf_I9_J,axiom,
    ! [D3: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S))) ) ) ).

% minf(9)
tff(fact_2466_minf_I10_J,axiom,
    ! [D3: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),Z2))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,S))) ) ) ).

% minf(10)
tff(fact_2467_minf_I10_J,axiom,
    ! [D3: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),S))) ) ) ).

% minf(10)
tff(fact_2468_minf_I10_J,axiom,
    ! [D3: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),S))) ) ) ).

% minf(10)
tff(fact_2469_minf_I10_J,axiom,
    ! [D3: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S))) ) ) ).

% minf(10)
tff(fact_2470_minf_I10_J,axiom,
    ! [D3: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S))) ) ) ).

% minf(10)
tff(fact_2471_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,aa_num_num(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_2472_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_2473_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_2474_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_2475_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_2476_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,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_2477_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_2478_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(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_2479_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(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_2480_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_2481_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_2482_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_2483_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_2484_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(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_2485_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_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_2486_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_2487_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% zle_iff_zadd
tff(fact_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_2494_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(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_2495_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),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,N)),one_one_rat)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_2496_numeral__Bit0__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(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit0_div_2
tff(fact_2497_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit0_div_2
tff(fact_2498_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_2499_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),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2500_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_2501_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(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2502_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_2503_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),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_strict_increasing
tff(fact_2504_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_2505_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(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2506_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_2507_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),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2508_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_2509_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(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2510_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_2511_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,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_2512_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_2513_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(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_2514_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_2515_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),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2516_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_2517_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(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2518_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_2519_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,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_2520_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_2521_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(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_2522_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_2523_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_2524_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(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2525_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_2526_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_2527_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_2528_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_2529_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_2530_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_2531_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_2532_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_2533_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_2534_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_2535_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_2536_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(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_2537_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_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_2538_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(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_2539_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_2540_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,aa_num_num(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,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_2541_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,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_2542_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_2543_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(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_2544_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,aa_num_num(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,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_2545_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,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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_2546_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
        & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( bit_se1745604003318907178nteger(N,A) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2547_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2548_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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,aa_num_num(bit0,one))),N)),one_one_nat) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2549_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) )
        & ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2550_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,aa_num_num(bit0,one))),A))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2551_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,aa_num_num(bit0,one))),A))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) )
        & ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2552_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,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_2553_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,aa_num_num(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,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_2554_power2__commute,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_2555_power2__commute,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_2556_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ) ).

% odd_even_add
tff(fact_2557_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,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% odd_even_add
tff(fact_2558_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,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% odd_even_add
tff(fact_2559_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_2560_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_2561_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),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(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_2562_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),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(aa_int_fun_int_int(plus_plus_int,X4),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(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_2563_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_2564_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),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(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))) ) ).

% floor_exists
tff(fact_2565_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se7879613467334960850it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se7879613467334960850it_int(M),A)) = aa_int_int(bit_se7879613467334960850it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_2566_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_2567_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se2159334234014336723it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se2159334234014336723it_int(M),A)) = aa_int_int(bit_se2159334234014336723it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_2568_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_2569_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se4203085406695923979it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se4203085406695923979it_int(M),A)) = aa_int_int(bit_se4203085406695923979it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_2570_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_2571_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_2572_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,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_2573_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_2574_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_2575_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(aa_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).

% le_imp_0_less
tff(fact_2576_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_2577_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_2578_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2579_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),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_2580_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_2581_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% half_gt_zero_iff
tff(fact_2582_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_2583_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,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_2584_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,aa_num_num(bit0,one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_2585_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,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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_2586_ceiling__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I4: 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,I4)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I4))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% ceiling_split
tff(fact_2587_ceiling__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim2889992004027027881ng_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4)),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,I4))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% ceiling_split
tff(fact_2588_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_2589_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_2590_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_2591_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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)))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_2592_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_2593_ceiling__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(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,aa_rat_int(archim2889992004027027881ng_rat,X)))) ) ).

% ceiling_correct
tff(fact_2594_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,aa_num_num(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_2595_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_2596_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_2597_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% le_ceiling_iff
tff(fact_2598_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_2599_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2600_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),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2601_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_2602_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2603_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2604_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2605_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_2606_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_2607_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_2608_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2609_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),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2610_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2611_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_2612_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_2613_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_2614_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_2615_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D3: 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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_2616_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_2617_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_2618_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_2619_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),aa_rat_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_2620_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_2621_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),aa_rat_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_2622_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_2623_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,aa_rat_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_2624_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_2625_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,aa_rat_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_2626_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_2627_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_2628_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_2629_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,aa_rat_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_2630_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_2631_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,aa_rat_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_2632_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_2633_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),aa_rat_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_2634_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_2635_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),aa_rat_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_2636_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_2637_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_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_2638_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,aa_num_num(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,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_2639_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_2640_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_2641_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_2642_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_2643_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_2644_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2645_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_2646_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_2647_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2648_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2649_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_2650_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_2651_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_2652_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_2653_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_2654_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_2655_diff__diff__left,axiom,
    ! [I: nat,J: 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),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_2656_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(6)
tff(fact_2657_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_2658_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_2659_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2660_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_2661_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_2662_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2663_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2664_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_2665_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2666_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_2667_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_2668_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_2669_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_2670_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2671_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_2672_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2673_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_2674_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2675_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_2676_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2677_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_2678_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),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)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_2679_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_2680_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_2681_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(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2682_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(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2683_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_2684_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_2685_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_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_2686_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_2687_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_2688_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,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_2689_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(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(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_2690_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(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_2691_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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).

% even_diff_nat
tff(fact_2692_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_2693_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_2694_diff__commute,axiom,
    ! [I: nat,J: 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),J)),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)),J) ).

% diff_commute
tff(fact_2695_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_2696_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_2697_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(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_2698_real__of__int__div,axiom,
    ! [D3: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),N))
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N),D3)) = 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,D3)) ) ) ).

% real_of_int_div
tff(fact_2699_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_2700_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A4)) )
     => ( ! [A4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),zero_zero_nat))
       => ( ! [A4: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_2701_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_2702_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_2703_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_2704_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_2705_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => 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),J))) ) ).

% trans_less_add2
tff(fact_2706_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_2707_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => 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,J),K))) ) ).

% add_less_mono1
tff(fact_2708_not__add__less2,axiom,
    ! [J: 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,J),I)),I)) ).

% not_add_less2
tff(fact_2709_not__add__less1,axiom,
    ! [I: nat,J: 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),J)),I)) ).

% not_add_less1
tff(fact_2710_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( 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,J),L))) ) ) ).

% add_less_mono
tff(fact_2711_add__lessD1,axiom,
    ! [I: nat,J: 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),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_2712_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_2713_less__diff__conv,axiom,
    ! [I: nat,J: 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,J),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)),J)) ) ).

% less_diff_conv
tff(fact_2714_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_2715_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_2716_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_2717_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_2718_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_2719_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_2720_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( 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,J),L))) ) ) ).

% add_le_mono
tff(fact_2721_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => 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,J),K))) ) ).

% add_le_mono1
tff(fact_2722_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => 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,J),M))) ) ).

% trans_le_add1
tff(fact_2723_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => 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),J))) ) ).

% trans_le_add2
tff(fact_2724_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_2725_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_2726_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_2727_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_2728_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( 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,J),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)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_2729_le__diff__conv,axiom,
    ! [J: 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,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_2730_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_2731_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_2732_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y))
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(Y),Z)) = aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_2733_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(divide_divide_int(N),X))))) ).

% real_of_int_div2
tff(fact_2734_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(divide_divide_int(N),X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_2735_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_2736_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [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) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2737_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)) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ pp(aa_nat_bool(P,D4)) ) ) ) ).

% nat_diff_split_asm
tff(fact_2738_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)) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => pp(aa_nat_bool(P,D4)) ) ) ) ).

% nat_diff_split
tff(fact_2739_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M4)),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_2740_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_2741_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_2742_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_2743_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(aa_int_fun_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_2744_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_2745_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),X_1)) ).

% linordered_field_no_ub
tff(fact_2746_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),X_1)) ).

% linordered_field_no_ub
tff(fact_2747_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X5)) ).

% linordered_field_no_lb
tff(fact_2748_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X5)) ).

% linordered_field_no_lb
tff(fact_2749_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_2750_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_2751_real__of__nat__div,axiom,
    ! [D3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D3)) ) ) ).

% real_of_nat_div
tff(fact_2752_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_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_2753_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,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_2754_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_2755_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(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_2756_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(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_2757_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_2758_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_2759_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_2760_nat__add__distrib,axiom,
    ! [Z: int,Z4: 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),Z4))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_2761_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,aa_num_num(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,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2762_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2763_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,aa_num_num(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,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2764_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2765_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,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2766_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2767_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,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_2768_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_2769_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_2770_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_2771_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_2772_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_2773_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,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_2774_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,aa_num_num(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_2775_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_2776_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,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_2777_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_2778_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2779_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_2780_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,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2781_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_2782_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,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2783_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_2784_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2785_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_2786_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),aa_rat_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_2787_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_2788_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,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_right_mono
tff(fact_2789_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_2790_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_2791_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_2792_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,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2793_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_2794_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,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_2795_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_2796_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_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_2797_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_2798_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_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,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_2799_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_2800_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_2801_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_2802_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_2803_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_2804_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,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_2805_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_2806_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,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_2807_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_2808_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_2809_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_2810_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_2811_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2812_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2813_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,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_2814_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_2815_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_2816_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_2817_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,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).

% frac_le
tff(fact_2818_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_2819_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,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).

% frac_less
tff(fact_2820_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_2821_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,aa_rat_rat(divide_divide_rat(X),Z)),aa_rat_rat(divide_divide_rat(Y),W))) ) ) ) ) ).

% frac_less2
tff(fact_2822_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_2823_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_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,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_2824_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_2825_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,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_2826_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_2827_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_2828_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_2829_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_2830_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_2831_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,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_2832_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_2833_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_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_2834_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_2835_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_2836_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_2837_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,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B)) ) ).

% gt_half_sum
tff(fact_2838_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_2839_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),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)))) ) ).

% less_half_sum
tff(fact_2840_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,aa_num_num(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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2841_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,aa_num_num(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(aa_int_fun_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_2842_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_2843_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),aa_rat_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_2844_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_2845_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_2846_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_2847_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_2848_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_2849_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(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_2850_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_2851__C5_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(4)
tff(fact_2852_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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_2853_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_2854_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_2855__C5_Ohyps_C_I3_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(3)
tff(fact_2856_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(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_2857_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(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_2858_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(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_2859_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(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_2860_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(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_2861_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(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_2862_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(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_2863_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(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_2864_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(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_2865_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(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_2866_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(divide_divide_int(K),L))) ) ) ).

% div_positive_int
tff(fact_2867_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(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_2868_zdiv__mono2__neg,axiom,
    ! [A: int,B2: 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),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B2)),aa_int_int(divide_divide_int(A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2869_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A2))
     => ( 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(divide_divide_int(A2),B)),aa_int_int(divide_divide_int(A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_2870_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_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_2871_zdiv__mono2,axiom,
    ! [A: int,B2: 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),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B2))) ) ) ) ).

% zdiv_mono2
tff(fact_2872_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A2))
     => ( 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(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A2),B))) ) ) ).

% zdiv_mono1
tff(fact_2873_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(divide_divide_int(X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_2874_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_2875_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(divide_divide_int(A),B))) ) ) ).

% div_positive
tff(fact_2876_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_2877_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(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2878_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_2879_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(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).

% discrete
tff(fact_2880_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,B3: real,C2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),C2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),B3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),C2)) ) ) ) )
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
               => ? [D: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
                    & ! [A4: real,B3: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),X4))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B3))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A4)),D)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B3)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_2881_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_2882_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(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)),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_2883_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,aa_num_num(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,aa_num_num(bit0,one))),M))) ) ).

% high_bound_aux
tff(fact_2884_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,aa_num_num(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,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_2885_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(aa_int_fun_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_2886__C5_Ohyps_C_I2_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "5.hyps"(2)
tff(fact_2887_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,aa_num_num(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_2888_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,aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_2889_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_2890_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_2891_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_2900_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_2901_floor__of__int,axiom,
    ! [Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_int_rat(ring_1_of_int_rat,Z)) = Z ).

% floor_of_int
tff(fact_2902_floor__of__int,axiom,
    ! [Z: int] : aa_real_int(archim6058952711729229775r_real,aa_int_real(ring_1_of_int_real,Z)) = Z ).

% floor_of_int
tff(fact_2903_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) = X )
    <=> ? [N3: int] : X = aa_int_rat(ring_1_of_int_rat,N3) ) ).

% of_int_floor_cancel
tff(fact_2904_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) = X )
    <=> ? [N3: int] : X = aa_int_real(ring_1_of_int_real,N3) ) ).

% of_int_floor_cancel
tff(fact_2905_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_2906_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_2907_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_2908_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% tanh_real_le_iff
tff(fact_2909_round__of__int,axiom,
    ! [N: int] : aa_real_int(archim8280529875227126926d_real,aa_int_real(ring_1_of_int_real,N)) = N ).

% round_of_int
tff(fact_2910_round__of__int,axiom,
    ! [N: int] : aa_rat_int(archim7778729529865785530nd_rat,aa_int_rat(ring_1_of_int_rat,N)) = N ).

% round_of_int
tff(fact_2911_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_2912_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_2913_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_2914_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_2915_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_2916_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_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_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_2924_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_2925_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_2926_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_2927_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_2928_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_2929_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_2930_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_2931_floor__zero,axiom,
    aa_rat_int(archim3151403230148437115or_rat,zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_2932_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_2933_floor__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_2934_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_2935_floor__one,axiom,
    aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).

% floor_one
tff(fact_2936_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_2937_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_2938_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_2939_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_2940_floor__of__nat,axiom,
    ! [N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% floor_of_nat
tff(fact_2941_floor__of__nat,axiom,
    ! [N: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% floor_of_nat
tff(fact_2942_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_2943_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_2944_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_2945_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_2946_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_2947_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_2948_round__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_2949_round__numeral,axiom,
    ! [N: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_2950_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_2951_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_2952_round__of__nat,axiom,
    ! [N: nat] : aa_real_int(archim8280529875227126926d_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% round_of_nat
tff(fact_2953_round__of__nat,axiom,
    ! [N: nat] : aa_rat_int(archim7778729529865785530nd_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% round_of_nat
tff(fact_2954_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_2955_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_2956_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_2957_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_2958_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_2959_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_2960_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( 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,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2961_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( 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,J),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,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2962_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_2963_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_2964_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_2965_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_2966_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_2967_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_2968_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,M)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_powr_numeral_real
tff(fact_2969_floor__diff__of__int,axiom,
    ! [X: rat,Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) ).

% floor_diff_of_int
tff(fact_2970_floor__diff__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) ).

% floor_diff_of_int
tff(fact_2971_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,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_2972_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_2973_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_2974_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,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_2975_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,aa_num_num(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,aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_2976_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_2977_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_2978_zero__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X)) ) ).

% zero_le_floor
tff(fact_2979_zero__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_floor
tff(fact_2980_floor__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat)) ) ).

% floor_less_zero
tff(fact_2981_floor__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% floor_less_zero
tff(fact_2982_numeral__le__floor,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)),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X)) ) ).

% numeral_le_floor
tff(fact_2983_numeral__le__floor,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(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X)) ) ).

% numeral_le_floor
tff(fact_2984_zero__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% zero_less_floor
tff(fact_2985_zero__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% zero_less_floor
tff(fact_2986_floor__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_le_zero
tff(fact_2987_floor__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_le_zero
tff(fact_2988_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_numeral
tff(fact_2989_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_numeral
tff(fact_2990_one__le__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).

% one_le_floor
tff(fact_2991_one__le__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).

% one_le_floor
tff(fact_2992_floor__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).

% floor_less_one
tff(fact_2993_floor__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% floor_less_one
tff(fact_2994_floor__diff__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_2995_floor__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_2996_floor__diff__one,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ).

% floor_diff_one
tff(fact_2997_floor__diff__one,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) ).

% floor_diff_one
tff(fact_2998_floor__numeral__power,axiom,
    ! [X: num,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% floor_numeral_power
tff(fact_2999_floor__numeral__power,axiom,
    ! [X: num,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% floor_numeral_power
tff(fact_3000_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_3001_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_3002_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_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_3003_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,aa_num_num(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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3004_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,aa_num_num(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,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3005_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_3006_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),aa_num_real(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_3007_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,aa_num_num(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_3008_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X)) ) ).

% numeral_less_floor
tff(fact_3009_numeral__less__floor,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(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),X)) ) ).

% numeral_less_floor
tff(fact_3010_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat))) ) ).

% floor_le_numeral
tff(fact_3011_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real))) ) ).

% floor_le_numeral
tff(fact_3012_one__less__floor,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_3013_one__less__floor,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) ) ).

% one_less_floor
tff(fact_3014_floor__le__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_3015_floor__le__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% floor_le_one
tff(fact_3016_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),aa_num_real(numeral_numeral_real,B))) = aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_3017_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_3018_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_3019_floor__le__round,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim8280529875227126926d_real,X))) ).

% floor_le_round
tff(fact_3020_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_3021_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_3022_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_3023_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_3024_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_3025_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_3026_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_3027_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_3028_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_3029_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_3030_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_3031_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_3032_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_3033_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_3034_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_3035_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_3036_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_3037_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_3038_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_3039_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_3040_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I3: nat] :
            ( ( J = 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),J))
             => ( 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_3041_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_3042_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_3043_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_3044_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_3045_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))
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M5)) ) ) ).

% Suc_less_eq2
tff(fact_3046_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_3047_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_3048_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_3049_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_3050_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_3051_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_3052_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_3053_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))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_3054_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_3055_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) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_3056_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_3057_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_3058_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_3059_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_3060_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_3061_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_3062_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M6))
     => ? [M4: nat] : M6 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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))
     => ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X4),X4))
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X4),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y3),Z2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X4),Z2)) ) )
         => ( ! [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_3069_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_3070_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_3071_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))) ) ).

% floor_mono
tff(fact_3072_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))) ) ).

% floor_mono
tff(fact_3073_of__int__floor__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,aa_rat_int(archim3151403230148437115or_rat,X))),X)) ).

% of_int_floor_le
tff(fact_3074_of__int__floor__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(archim6058952711729229775r_real,X))),X)) ).

% of_int_floor_le
tff(fact_3075_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% floor_less_cancel
tff(fact_3076_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% floor_less_cancel
tff(fact_3077_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_3078_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_3079_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_3080_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_3081_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_3082_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_3083_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_3084_powr__mono,axiom,
    ! [A: real,B: real,X: 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,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))) ) ) ).

% powr_mono
tff(fact_3085_floor__le__ceiling,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim7802044766580827645g_real,X))) ).

% floor_le_ceiling
tff(fact_3086_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_3087_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_3088_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_3089_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_3090_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_3091_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N5: 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),N5))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N5))) ) ) ).

% lift_Suc_mono_less
tff(fact_3092_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N5: 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),N5))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N5))) ) ) ).

% lift_Suc_mono_less
tff(fact_3093_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N5: 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),N5))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N5))) ) ) ).

% lift_Suc_mono_less
tff(fact_3094_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N5: 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),N5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N5))) ) ) ).

% lift_Suc_mono_less
tff(fact_3095_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N5: 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),N5))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N5))) ) ) ).

% lift_Suc_mono_less
tff(fact_3096_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N5: 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),N5))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N5))) ) ) ).

% lift_Suc_mono_le
tff(fact_3097_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N5: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_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_eq_nat,N),N5))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N5))) ) ) ).

% lift_Suc_mono_le
tff(fact_3098_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N5: 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),N5))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N5))) ) ) ).

% lift_Suc_mono_le
tff(fact_3099_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N5: 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),N5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N5))) ) ) ).

% lift_Suc_mono_le
tff(fact_3100_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N5: 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),N5))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N5))) ) ) ).

% lift_Suc_mono_le
tff(fact_3101_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N5: 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),N5))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N5)),aa_nat_set_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_3102_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N5: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N5)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_3103_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N5: 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),N5))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N5)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_3104_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N5: 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),N5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N5)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_3105_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N5: 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),N5))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N5)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_3106_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_3107_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_3108_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_3109_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_3110_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_3111_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_3112_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M4: nat] : N = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_3113_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_3114_gr0__conv__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_conv_Suc
tff(fact_3115_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_3116_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_3117_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_3118_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q4: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ) ).

% less_natE
tff(fact_3119_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_3120_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_3121_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_3122_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_3123_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_3124_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_3125_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( 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),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_3126_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [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),J))
               => ( 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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3133_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_3134_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_3135_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_3136_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_3137_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_3138_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_3139_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_3140_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_3141_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_3142_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% le_floor_iff
tff(fact_3143_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% le_floor_iff
tff(fact_3144_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% floor_less_iff
tff(fact_3145_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).

% floor_less_iff
tff(fact_3146_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_3147_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_3148_le__floor__add,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)))) ).

% le_floor_add
tff(fact_3149_le__floor__add,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))) ).

% le_floor_add
tff(fact_3150_int__add__floor,axiom,
    ! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),X)) ).

% int_add_floor
tff(fact_3151_int__add__floor,axiom,
    ! [Z: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).

% int_add_floor
tff(fact_3152_floor__add__int,axiom,
    ! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_3153_floor__add__int,axiom,
    ! [X: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_3154_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_3155_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_3156_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_3157_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_3158_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_3159_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_3160_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(divide_divide_int(K),L) ).

% floor_divide_of_int_eq
tff(fact_3161_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,K)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(divide_divide_int(K),L) ).

% floor_divide_of_int_eq
tff(fact_3162_floor__power,axiom,
    ! [X: rat,N: nat] :
      ( ( X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(X),N)) = aa_nat_int(power_power_int(aa_rat_int(archim3151403230148437115or_rat,X)),N) ) ) ).

% floor_power
tff(fact_3163_floor__power,axiom,
    ! [X: real,N: nat] :
      ( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,X)),N) ) ) ).

% floor_power
tff(fact_3164_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_3165_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y))) ) ).

% round_mono
tff(fact_3166_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_3167_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_3168_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_3169_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_3170_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_3171_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_3172_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_3173_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_3174_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_3175_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_3176_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_3177_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_3178_round__def,axiom,
    ! [X: rat] : aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_3179_round__def,axiom,
    ! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_3180_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_3181_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))
            & ! [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),K2))
               => ~ pp(aa_nat_bool(P,I2)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_3182_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_3183_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_3184_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_3185_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_3186_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_3187_ceiling__ge__round,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim7802044766580827645g_real,X))) ).

% ceiling_ge_round
tff(fact_3188_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_3189_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_3190_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
    <=> ? [N3: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_3191_of__nat__floor,axiom,
    ! [R2: rat] :
      ( 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,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R2)))),R2)) ) ).

% of_nat_floor
tff(fact_3192_of__nat__floor,axiom,
    ! [R2: 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_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2)) ) ).

% of_nat_floor
tff(fact_3193_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) ).

% one_add_floor
tff(fact_3194_one__add__floor,axiom,
    ! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% one_add_floor
tff(fact_3195_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(aa_int_fun_int_int(plus_plus_int,K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3196_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% floor_divide_of_nat_eq
tff(fact_3197_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% floor_divide_of_nat_eq
tff(fact_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_ceiling__altdef,axiom,
    ! [X: rat] :
      ( ( ( X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) )
      & ( ( X != aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ) ) ) ).

% ceiling_altdef
tff(fact_3206_ceiling__altdef,axiom,
    ! [X: real] :
      ( ( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) )
      & ( ( X != aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) ) ) ) ).

% ceiling_altdef
tff(fact_3207_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim6058952711729229775r_real,X))),one_one_int)) ).

% ceiling_diff_floor_le_1
tff(fact_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_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_3218_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_3219_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_3220_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_3221_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_Suc__as__int,axiom,
    ! [X5: nat] : aa_nat_nat(suc,X5) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),one_one_int)) ).

% Suc_as_int
tff(fact_3229_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z ) ) ) ).

% floor_unique
tff(fact_3230_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),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,Z)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Z ) ) ) ).

% floor_unique
tff(fact_3231_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat))) ) ) ).

% floor_eq_iff
tff(fact_3232_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),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,A)),one_one_real))) ) ) ).

% floor_eq_iff
tff(fact_3233_floor__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,aa_rat_int(archim3151403230148437115or_rat,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I4)),one_one_rat))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% floor_split
tff(fact_3234_floor__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim6058952711729229775r_real,T)))
    <=> ! [I4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real))) )
         => pp(aa_int_bool(P,I4)) ) ) ).

% floor_split
tff(fact_3235_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X)) ) ).

% less_floor_iff
tff(fact_3236_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
    <=> 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,Z)),one_one_real)),X)) ) ).

% less_floor_iff
tff(fact_3237_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))) ) ).

% floor_le_iff
tff(fact_3238_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
    <=> 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,Z)),one_one_real))) ) ).

% floor_le_iff
tff(fact_3239_floor__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),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(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_3240_floor__correct,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(archim6058952711729229775r_real,X))),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)))) ) ).

% floor_correct
tff(fact_3241_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_3242_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_3243_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_3244_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_3245_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_3246_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(divide_divide_int(aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_3247_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [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)))),E2)) ) ).

% nat_approx_posE
tff(fact_3248_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E2)) ) ).

% nat_approx_posE
tff(fact_3249_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,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_3250_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,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_3251_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_3252_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_3253_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(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_3254_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_3255_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_3256_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,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_3257_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,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_3258_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_3259_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(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_3260_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,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_3261_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,aa_num_num(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_3262_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_3263_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,aa_rat_int(archim7778729529865785530nd_rat,X))),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))) ).

% of_int_round_le
tff(fact_3264_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_ge
tff(fact_3265_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_ge
tff(fact_3266_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_gt
tff(fact_3267_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X)))) ).

% of_int_round_gt
tff(fact_3268_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,aa_num_num(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,aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3269_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,aa_num_num(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,aa_num_num(bit0,one))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_3270_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,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_3271_round__altdef,axiom,
    ! [X: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim2889992004027027881ng_rat,X) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,X) ) ) ) ).

% round_altdef
tff(fact_3272_round__altdef,axiom,
    ! [X: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim7802044766580827645g_real,X) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)))
       => ( aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,X) ) ) ) ).

% round_altdef
tff(fact_3273_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,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_3274_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_3275_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_3276_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_3277_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_3278_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_3279_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_3280_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_3281_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_3282_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_3283_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_3284_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_3285_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_3286_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_3287_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_3288_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_3289_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_3290_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_3291_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_3292_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_3293_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_3294_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_3295_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_3296_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_3297_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_3298_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_3299_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_3300_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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_3301_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_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_3302_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_3303_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_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_3304_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_3305_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_3306_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_3307_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_3308_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_3309_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_3310_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_3311_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_3312_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_3313_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_3314_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_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_3315_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_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_3316_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_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_3317_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_bit__concat__def,axiom,
    ! [H: nat,L: nat,D3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D3) = 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,aa_num_num(bit0,one))),D3))),L) ).

% bit_concat_def
tff(fact_3325_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_3326_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_3327_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_3328_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_3329_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_3330_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_3331_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_3332_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_3333_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_3334_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_3335_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_3336_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_3337_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_3338_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_3339_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_3340_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_3341_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_3342_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_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_3343_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_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_3344_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_3345_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_3346_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_3347_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_3348_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_3349_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_3350_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_3351_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_3352_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3353_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_3354_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3355_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_3356_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_3357_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3358_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_3359_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3360_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_3361_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_3362_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3363_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_3364_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_3365_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3366_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_3367_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_3368_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3369_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_3370_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_3371_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3372_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_3373_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_3374_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( aa_rat_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 )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3375_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_3376_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)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_3377_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_3378_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_3379_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_3380_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_3381_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_3382_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_3383_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_3384_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_3385_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_3386_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3387_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3388_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3389_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_3390_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)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_3391_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_3392_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_3393_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_3394_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_3395_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(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_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_3396_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_3397_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_3398_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(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_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_3399_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_3400_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_3401_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_3402_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_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_3403_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_3404_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_3405_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_3406_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3407_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_3408_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_3409_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_3410_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_3411_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_3412_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_3413_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(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_3414_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_3415_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_3416_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_3417_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_3418_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_3419_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_3420_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_3421_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_3422_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),aa_num_real(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),aa_num_real(numeral_numeral_real,W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3423_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),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3424_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),aa_num_real(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),aa_num_real(numeral_numeral_real,W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3425_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,aa_rat_rat(divide_divide_rat(B),aa_num_rat(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),aa_num_rat(numeral_numeral_rat,W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3426_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_3427_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3428_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3429_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_3430_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) = B ) )
        & ( ( aa_num_real(numeral_numeral_real,W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3431_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3432_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),aa_num_real(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),aa_num_real(numeral_numeral_real,W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3433_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,aa_rat_rat(divide_divide_rat(B),aa_num_rat(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),aa_num_rat(numeral_numeral_rat,W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3434_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),aa_num_real(numeral_numeral_real,W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3435_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),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3436_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_3437_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_3438_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_3439_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_3440_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_3441_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_3442_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_3443_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_3444_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_3445_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_3446_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_3447_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_3448_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_3449_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_3450_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_3451_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_3452_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_3453_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_3454_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_3455_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_3456_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_3457_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_3458_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_3459_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] : aa_int_int(aa_int_fun_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_3460_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_3461_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_3462_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_3463_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_3464_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_int_int(aa_int_fun_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_3465_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).

% even_mult_iff
tff(fact_3466_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,aa_num_num(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,aa_num_num(bit0,one))),A))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_mult_iff
tff(fact_3467_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),A))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_mult_iff
tff(fact_3468_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% or_numerals(3)
tff(fact_3469_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% or_numerals(3)
tff(fact_3470_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% xor_numerals(3)
tff(fact_3471_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% xor_numerals(3)
tff(fact_3472_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3473_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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3474_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3475_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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3476_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,aa_num_num(bit0,one))),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3477_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(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_3478_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3479_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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3480_ab__semigroup__mult__class_Omult__ac_I1_J,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)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_3481_ab__semigroup__mult__class_Omult__ac_I1_J,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)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_3482_ab__semigroup__mult__class_Omult__ac_I1_J,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)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_3483_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_3484_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_3485_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_3486_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_3487_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_3488_mult_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) ).

% mult.commute
tff(fact_3489_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_3490_mult_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) ).

% mult.commute
tff(fact_3491_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% mult.commute
tff(fact_3492_mult_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)) ).

% mult.left_commute
tff(fact_3493_mult_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)) ).

% mult.left_commute
tff(fact_3494_mult_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)) ).

% mult.left_commute
tff(fact_3495_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_3496_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_3497_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_3498_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_3499_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3500_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_3501_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_3502_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_3503_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3504_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_3505_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_3506_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_3507_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_3508_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_3509_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_3510_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_3511_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_3512_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_3513_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_3514_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_3515_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_3516_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3517_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_3518_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_3519_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_3520_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_3521_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_3522_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_3523_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_3524_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_3525_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_3526_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D5: real,Q: fun_real_bool] :
      ( ! [X4: real,K2: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D5)))) )
     => ( ! [X4: real,K2: real] :
            ( pp(aa_real_bool(Q,X4))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D5)))) )
       => ! [X5: real,K4: real] :
            ( ( pp(aa_real_bool(P,X5))
              | pp(aa_real_bool(Q,X5)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))))
              | pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3527_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D5: rat,Q: fun_rat_bool] :
      ( ! [X4: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X4))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D5)))) )
     => ( ! [X4: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X4))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D5)))) )
       => ! [X5: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X5))
              | pp(aa_rat_bool(Q,X5)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X5),aa_rat_rat(times_times_rat(K4),D5))))
              | pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X5),aa_rat_rat(times_times_rat(K4),D5)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3528_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D5: int,Q: fun_int_bool] :
      ( ! [X4: int,K2: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5)))) )
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(Q,X4))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5)))) )
       => ! [X5: int,K4: int] :
            ( ( pp(aa_int_bool(P,X5))
              | pp(aa_int_bool(Q,X5)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3529_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D5: real,Q: fun_real_bool] :
      ( ! [X4: real,K2: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D5)))) )
     => ( ! [X4: real,K2: real] :
            ( pp(aa_real_bool(Q,X4))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D5)))) )
       => ! [X5: real,K4: real] :
            ( ( pp(aa_real_bool(P,X5))
              & pp(aa_real_bool(Q,X5)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))))
              & pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3530_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D5: rat,Q: fun_rat_bool] :
      ( ! [X4: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X4))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D5)))) )
     => ( ! [X4: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X4))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D5)))) )
       => ! [X5: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X5))
              & pp(aa_rat_bool(Q,X5)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X5),aa_rat_rat(times_times_rat(K4),D5))))
              & pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X5),aa_rat_rat(times_times_rat(K4),D5)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3531_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D5: int,Q: fun_int_bool] :
      ( ! [X4: int,K2: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5)))) )
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(Q,X4))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5)))) )
       => ! [X5: int,K4: int] :
            ( ( pp(aa_int_bool(P,X5))
              & pp(aa_int_bool(Q,X5)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3532_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_3533_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_3534_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_3535_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_3536_power__commutes,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_commutes
tff(fact_3537_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_3538_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_3539_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_3540_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_3541_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_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_3542_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_3543_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_3544_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_3545_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_3546_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_3547_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_3548_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_3549_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_3550_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_3551_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_3552_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_3553_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_3554_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_3555_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_3556_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_3557_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_3558_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_3559_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_3560_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_3561_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: 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,J),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),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_3562_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => 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),J))) ) ).

% mult_le_mono2
tff(fact_3563_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => 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,J),K))) ) ).

% mult_le_mono1
tff(fact_3564_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( 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,J),L))) ) ) ).

% mult_le_mono
tff(fact_3565_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_3566_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_3567_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_3568_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_3569_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_3570_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_3571_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_3572_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_3573_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_3574_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_3575_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : 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_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,A))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))))) ).

% le_mult_nat_floor
tff(fact_3576_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : 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_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))))) ).

% le_mult_nat_floor
tff(fact_3577_le__mult__floor,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,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor
tff(fact_3578_le__mult__floor,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_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor
tff(fact_3579_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(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le
tff(fact_3580_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,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le
tff(fact_3581_mult__mono,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_mono
tff(fact_3582_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_mono
tff(fact_3583_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_mono
tff(fact_3584_mult__mono,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => ( 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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))) ) ) ) ) ).

% mult_mono
tff(fact_3585_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_mono'
tff(fact_3586_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_mono'
tff(fact_3587_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_mono'
tff(fact_3588_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => ( 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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))) ) ) ) ) ).

% mult_mono'
tff(fact_3589_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_3590_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_3591_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(aa_int_fun_int_int(times_times_int,A),A))) ).

% zero_le_square
tff(fact_3592_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_3593_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_3594_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(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% split_mult_pos_le
tff(fact_3595_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_3596_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_3597_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_3598_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_3599_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_3600_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(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_3601_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_3602_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_3603_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_3604_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono
tff(fact_3605_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_3606_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_3607_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_3608_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_3609_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_3610_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_3611_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono
tff(fact_3612_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_3613_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_3614_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(aa_int_fun_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_3615_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_3616_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_3617_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_3618_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(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_3619_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_3620_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_3621_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_3622_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(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_3623_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_3624_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_3625_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_3626_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(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_3627_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_3628_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_3629_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_3630_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(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_3631_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_3632_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_3633_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_3634_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(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3635_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_3636_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_3637_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(aa_int_fun_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_3638_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_3639_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_3640_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_3641_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3642_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_3643_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_3644_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(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_neg_neg
tff(fact_3645_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_3646_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_3647_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_3648_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_3649_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_3650_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_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_3651_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_3652_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_3653_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_3654_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(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_3655_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_3656_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_3657_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_3658_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(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_3659_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_3660_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_3661_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_3662_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(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_pos_pos
tff(fact_3663_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_3664_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_3665_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_3666_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(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_3667_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_3668_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_3669_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(aa_int_fun_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_3670_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_3671_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_3672_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_3673_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(aa_int_fun_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_3674_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_3675_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_3676_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_3677_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(aa_int_fun_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_3678_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_3679_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_3680_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3681_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_3682_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_3683_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3684_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_3685_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_3686_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3687_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_3688_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_3689_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_3690_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_3691_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_3692_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_3693_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3694_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_3695_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_3696_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3697_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_3698_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_3699_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_3700_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_3701_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_3702_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_3703_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_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_3704_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_3705_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_3706_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_3707_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3708_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_3709_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_3710_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_3711_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(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).

% less_1_mult
tff(fact_3712_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_3713_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_3714_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3715_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_3716_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_3717_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3718_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_3719_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_3720_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_3721_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_3722_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_3723_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_3724_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_3725_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_3726_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(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_3727_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_3728_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_3729_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_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_3730_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_3731_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_3732_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(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_3733_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_3734_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_3735_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_3736_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_3737_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_3738_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_3739_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_3740_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_3741_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_3742_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_3743_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_3744_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),A)),B)) ).

% mult_diff_mult
tff(fact_3745_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B)) ).

% mult_diff_mult
tff(fact_3746_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_3747_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_3748_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_3749_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_3750_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_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_3751_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_3752_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_3753_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_3754_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_3755_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_3756_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_3757_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_3758_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_3759_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_3760_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_3761_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_3762_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_3763_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_3764_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_3765_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_3766_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_3767_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_3768_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(aa_int_fun_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_3769_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( 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,J),K))) ) ) ).

% mult_less_mono1
tff(fact_3770_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( 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),J))) ) ) ).

% mult_less_mono2
tff(fact_3771_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_3772_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_3773_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_3774_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_3775_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_3776_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_3777_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_3778_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_3779_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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3780_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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3781_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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3782_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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3783_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,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3784_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_3785_frac__ge__0,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_ge_0
tff(fact_3786_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_3787_frac__lt__1,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).

% frac_lt_1
tff(fact_3788_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_3789_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_3790_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3791_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3792_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3793_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => ( 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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3794_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3795_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3796_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3797_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => ( 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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3798_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_3799_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_3800_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_3801_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_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_3802_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_3803_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_3804_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_3805_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3806_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_3807_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_3808_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3809_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_3810_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_3811_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3812_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_3813_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_3814_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_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_3815_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3816_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3817_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3818_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => ( 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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3819_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_3820_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_3821_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_3822_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_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_3823_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_3824_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_3825_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3826_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3827_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3828_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3))
       => ( 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),D3))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3829_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D3: 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),D3))
       => ( 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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))) ) ) ) ) ).

% mult_strict_mono
tff(fact_3830_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_3831_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_3832_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_3833_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3834_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_3835_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_3836_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(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_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_3837_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_3838_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_3839_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(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_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_3840_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_3841_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_3842_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(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_3843_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_3844_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_3845_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(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_3846_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_3847_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_3848_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_3849_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(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_3850_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_3851_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_3852_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_3853_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(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_3854_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_3855_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_3856_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_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_3857_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_3858_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),aa_rat_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_3859_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3860_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_3861_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),aa_rat_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_3862_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3863_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_3864_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_3865_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_3866_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_3867_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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3868_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_3869_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_3870_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_3871_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_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)),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_3872_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_3873_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,aa_rat_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_3874_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_3875_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),aa_rat_rat(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_3876_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_3877_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,aa_rat_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_3878_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_3879_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),aa_rat_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)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_3880_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_3881_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,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_3882_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_3883_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_3884_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_3885_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_3886_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_3887_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3888_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_3889_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3890_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3891_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_3892_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_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_num_real(numeral_numeral_real,W)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3893_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3894_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D3: 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),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D3)))
    <=> 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)),E2)),D3))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3895_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D3))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3896_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D3))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3897_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D3: 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),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D3)))
    <=> 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)),E2)),C)),D3)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3898_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D3)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3899_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D3)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3900_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_3901_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_3902_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3903_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_3904_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_3905_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3906_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_3907_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_3908_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3909_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_3910_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_3911_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3912_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_3913_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_3914_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_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_3915_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_3916_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_3917_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_3918_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_3919_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_3920_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_3921_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D3: 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),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D3)))
    <=> 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)),E2)),C)),D3)) ) ).

% less_add_iff1
tff(fact_3922_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D3)) ) ).

% less_add_iff1
tff(fact_3923_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D3)) ) ).

% less_add_iff1
tff(fact_3924_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D3: 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),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D3)))
    <=> 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)),E2)),D3))) ) ).

% less_add_iff2
tff(fact_3925_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D3)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D3))) ) ).

% less_add_iff2
tff(fact_3926_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D3))) ) ).

% less_add_iff2
tff(fact_3927_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_3928_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_3929_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3930_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_3931_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_3932_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3933_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_3934_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_3935_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_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_3936_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_3937_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_3938_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3939_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_3940_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_3941_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_3942_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(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3943_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_3944_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_3945_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_3946_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(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3947_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_3948_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_3949_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_3950_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_3951_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(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_3952_unity__coeff__ex,axiom,
    ! [P: fun_Co8822979462182099027r_bool,L: code_integer] :
      ( ? [X3: code_integer] : pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3)))
    <=> ? [X3: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X3,zero_z3403309356797280102nteger)))
          & pp(aa_Code_integer_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3953_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X3: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X3)))
    <=> ? [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),zero_zero_real)))
          & pp(aa_real_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3954_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X3: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X3)))
    <=> ? [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X3),zero_zero_rat)))
          & pp(aa_rat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3955_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X3: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3)))
    <=> ? [X3: 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,X3),zero_zero_nat)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3956_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X3: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3)))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int)))
          & pp(aa_int_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_3957_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D3: 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),D3))
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D3,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D3) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3958_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D3: 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),D3))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D3),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),D3) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3959_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D3: 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),D3))
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D3),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D3) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3960_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3961_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_3962_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(divide_divide_int(B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3963_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3964_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_3965_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(divide_divide_int(A),B)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3966_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3967_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_3968_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(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3969_inf__period_I4_J,axiom,
    ! [D3: code_integer,D5: code_integer,T: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),D5))
     => ! [X5: code_integer,K4: code_integer] :
          ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,T)))
        <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X5,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D5)),T))) ) ) ).

% inf_period(4)
tff(fact_3970_inf__period_I4_J,axiom,
    ! [D3: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),D5))
     => ! [X5: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_3971_inf__period_I4_J,axiom,
    ! [D3: rat,D5: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),D5))
     => ! [X5: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X5),aa_rat_rat(times_times_rat(K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_3972_inf__period_I4_J,axiom,
    ! [D3: int,D5: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),D5))
     => ! [X5: int,K4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(4)
tff(fact_3973_inf__period_I3_J,axiom,
    ! [D3: code_integer,D5: code_integer,T: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),D5))
     => ! [X5: code_integer,K4: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(X5,T)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D3),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X5,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D5)),T))) ) ) ).

% inf_period(3)
tff(fact_3974_inf__period_I3_J,axiom,
    ! [D3: real,D5: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),D5))
     => ! [X5: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_3975_inf__period_I3_J,axiom,
    ! [D3: rat,D5: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),D5))
     => ! [X5: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(X5),T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D3),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X5),aa_rat_rat(times_times_rat(K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_3976_inf__period_I3_J,axiom,
    ! [D3: int,D5: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),D5))
     => ! [X5: int,K4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D5))),T))) ) ) ).

% inf_period(3)
tff(fact_3977_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_3978_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_3979_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_3980_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_3981_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_3982_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),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),Q3))) ) ) ).

% div_less_iff_less_mult
tff(fact_3983_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_3984_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_3985_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),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,J),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),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_3986_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( 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,J),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,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3987_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),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,J),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),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_3988_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( 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,J),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,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_3989_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),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,J),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),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_3990_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( 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,J),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,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_3991_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D6: nat,X4: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D6),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D6),B))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D6) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3992_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_3993_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_3994_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(aa_int_fun_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_3995_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_3996_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_3997_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(aa_int_fun_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_3998_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_3999_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_4000_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(aa_int_fun_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_4001_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_4002_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_4003_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(aa_int_fun_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_4004_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_4005_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_4006_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(aa_int_fun_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_4007_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_4008_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_4009_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_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_4010_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_4011_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_4012_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(aa_int_fun_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_4013_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_4014_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_4015_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_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_4016_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_4017_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_4018_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_4019_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_4020_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_4021_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_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)),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_4022_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_4023_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono
tff(fact_4024_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_4025_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,aa_rat_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_4026_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_4027_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),aa_rat_rat(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_4028_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_4029_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,aa_rat_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_4030_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_4031_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),aa_rat_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)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_4032_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_4033_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,aa_rat_rat(divide_divide_rat(X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_4034_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_4035_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),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_4036_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_4037_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_4038_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_4039_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))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_4040_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4041_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)),aa_num_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_num_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_num_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_num_real(numeral_numeral_real,W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4042_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,aa_rat_rat(divide_divide_rat(B),C)),aa_num_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(aa_num_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(aa_num_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),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4043_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_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_num_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_num_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_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4044_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,aa_num_rat(numeral_numeral_rat,W)),aa_rat_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(aa_num_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(aa_num_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,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4045_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_4046_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,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_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_4047_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_4048_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_4049_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_4050_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(aa_int_fun_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_4051_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_4052_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,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W),Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_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_4053_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_4054_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_4055_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_4056_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_4057_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_4058_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_4059_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_4060_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_4061_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_4062_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_4063_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(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4064_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4065_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4066_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,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4067_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_4068_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4069_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_4070_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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4071_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4072_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_4073_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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4074_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 )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B3),one_one_Code_integer))
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4075_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 )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B3),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = 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),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4076_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 )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),one_one_int))
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4077_evenE,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_4078_evenE,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_4079_evenE,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_4080_power4__eq__xxxx,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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_4081_power4__eq__xxxx,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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_4082_power4__eq__xxxx,axiom,
    ! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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_4083_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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_4084_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_4085_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_4086_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_4087_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_4088_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_4089_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_4090_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,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_double_not_eq_double
tff(fact_4091_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,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% double_not_eq_Suc_double
tff(fact_4092_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,aa_num_num(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,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_4093_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,aa_num_num(bit0,one))),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_4094_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,aa_num_num(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,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_4095_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,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_4096_frac__def,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,X) = aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))) ).

% frac_def
tff(fact_4097_frac__def,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))) ).

% frac_def
tff(fact_4098_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_4099_div__nat__eqI,axiom,
    ! [N: nat,Q3: 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),Q3)),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,Q3))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_4100_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 )
         => ! [I4: 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),I4)),J3) )
               => pp(aa_nat_bool(P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_4101_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_4102_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_4103_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q3))
     => ( 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),Q3)))
      <=> 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),Q3)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_4104_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( 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,J),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,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_4105_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),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,J),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),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_4106_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_4107_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_4108_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_4109_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q3),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),Q3)))
        <=> 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)),Q3)))) ) ) ) ).

% dvd_minus_add
tff(fact_4110_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_4111_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))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_4112_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4113_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_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_num_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_num_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_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4114_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,aa_num_rat(numeral_numeral_rat,W)),aa_rat_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(aa_num_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(aa_num_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,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4115_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)),aa_num_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_num_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_num_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_num_real(numeral_numeral_real,W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4116_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,aa_rat_rat(divide_divide_rat(B),C)),aa_num_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(aa_num_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(aa_num_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),aa_num_rat(numeral_numeral_rat,W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4117_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_4118_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,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4119_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( 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)) ) ) ).

% frac_eq
tff(fact_4120_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( 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),one_one_rat)) ) ) ).

% frac_eq
tff(fact_4121_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4122_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,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4123_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,aa_num_num(bit0,one))),A))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4124_frac__add,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(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_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(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_4125_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_4126_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_4127_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_4128_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_4129_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_4130_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_4131_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_4132_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(aa_int_fun_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_4133_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_4134_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_4135_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_4136_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_4137_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(aa_int_fun_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_4138_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)) )
        | ? [Q5: 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),Q5)),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,Q5))))
            & pp(aa_nat_bool(P,Q5)) ) ) ) ).

% split_div'
tff(fact_4139_oddE,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_4140_oddE,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ~ ! [B3: 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,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_4141_oddE,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_4142_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,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_4143_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,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_4144_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_4145_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,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_4146_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_4147_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,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_4148_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,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_4149_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,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_4150_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,aa_num_num(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,aa_num_num(bit0,one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_4151_floor__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P3),Q3)))),Q3)),P3)) ) ).

% floor_divide_lower
tff(fact_4152_floor__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q3)))),Q3)),P3)) ) ).

% floor_divide_lower
tff(fact_4153_ceiling__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => 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),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_4154_ceiling__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => 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,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q3)))),Q3))) ) ).

% ceiling_divide_upper
tff(fact_4155_vebt__buildup_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_buildup.cases
tff(fact_4156_option_Osize_I4_J,axiom,
    ! [X2: nat] : aa_option_nat_nat(size_size_option_nat,some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4157_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4158_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4159_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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_4160_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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_4161_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,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4162_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,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4163_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4164_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_4165_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,aa_num_num(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_4166_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,aa_num_num(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_4167_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,aa_num_num(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_4168_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,aa_num_num(bit0,one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_4169_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,aa_num_num(bit0,one))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_4170_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,aa_num_num(bit0,one))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_4171_floor__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P3),Q3)))),one_one_rat)),Q3))) ) ).

% floor_divide_upper
tff(fact_4172_floor__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q3)))),one_one_real)),Q3))) ) ).

% floor_divide_upper
tff(fact_4173_ceiling__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q3))
     => 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),Q3)))),one_one_real)),Q3)),P3)) ) ).

% ceiling_divide_lower
tff(fact_4174_ceiling__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q3)))),one_one_rat)),Q3)),P3)) ) ).

% ceiling_divide_lower
tff(fact_4175_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_4176_floor__add,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(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int) ) ) ) ).

% floor_add
tff(fact_4177_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_4178_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,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_4179_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(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),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_4180_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(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(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(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4181_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,aa_num_num(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,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4182_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,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4183_of__real__of__int__eq,axiom,
    ! [Z: int] : aa_real_real(real_V1803761363581548252l_real,aa_int_real(ring_1_of_int_real,Z)) = aa_int_real(ring_1_of_int_real,Z) ).

% of_real_of_int_eq
tff(fact_4184_of__real__of__int__eq,axiom,
    ! [Z: int] : aa_real_complex(real_V4546457046886955230omplex,aa_int_real(ring_1_of_int_real,Z)) = aa_int_complex(ring_17405671764205052669omplex,Z) ).

% of_real_of_int_eq
tff(fact_4185_of__real__of__nat__eq,axiom,
    ! [N: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ).

% of_real_of_nat_eq
tff(fact_4186_of__real__of__nat__eq,axiom,
    ! [N: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% of_real_of_nat_eq
tff(fact_4187_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_4188_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_4189_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% of_real_numeral
tff(fact_4190_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_4191_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_4192_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_4193_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_4194_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_4195_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_4196_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_4197_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_4198_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_4199_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% semiring_norm(13)
tff(fact_4200_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_4201_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_4202_num__double,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_4203_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_4204_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_4205_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_4206_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_4207_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa: 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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_4208_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_4209_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_4210_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_4211_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_4212_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_4213_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_4214_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_4215_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_4216_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_4217_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_4218_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_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_4219_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( 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(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_4220_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(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_4221_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(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_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_4222_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(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),T))) ) ) ).

% zdvd_mono
tff(fact_4223_zdvd__period,axiom,
    ! [A: int,D3: int,X: int,T: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D3))),T))) ) ) ).

% zdvd_period
tff(fact_4224_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(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_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_4225_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_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_4226_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_4227_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(aa_int_fun_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_4228_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_4229_minusinfinity,axiom,
    ! [D3: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P1,X4))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P1,X4)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P1,X_13))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_4230_plusinfinity,axiom,
    ! [D3: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P2,X4)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_4231_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(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_4232_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_4233_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_4234_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_4235_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( 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(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_4236_incr__mult__lemma,axiom,
    ! [D3: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X5: int] :
              ( pp(aa_int_bool(P,X5))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_4237_q__pos__lemma,axiom,
    ! [B2: int,Q6: int,R4: 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(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_4238_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
         => ( 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),B2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q3),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_4239_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q3)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_4240_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),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),Q3)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_4241_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),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,Q3),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_4242_nat__mult__distrib,axiom,
    ! [Z: int,Z4: 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(aa_int_fun_int_int(times_times_int,Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_4243_decr__mult__lemma,axiom,
    ! [D3: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X5: int] :
              ( pp(aa_int_bool(P,X5))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4244_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_4245_four__x__squared,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_4246_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))
       => ( ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M4))
             => 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,M4)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_4247_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_4248_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_4249_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_4250_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),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(divide_divide_int(A),B) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_4251_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),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(divide_divide_int(A),B) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_4252_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_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))
         => ! [I4: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D3: 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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D3))),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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D3),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_4260_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa: 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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_4261_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_4262_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_4263_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_4264_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,aa_num_num(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_4265_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_4266_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_4267_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: 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)),K5))
       => ( 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))),K5))
         => 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,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4268_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5))
         => 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,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4269_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,aa_num_num(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,aa_num_num(bit0,one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_4270_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_4271_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_4272_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(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_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_4273_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_4274_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_4275_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(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_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_4276_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,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4277_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(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_4278_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_4279_bit__split__inv,axiom,
    ! [X: nat,D3: 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),D3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D3)),D3) = X ).

% bit_split_inv
tff(fact_4280_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_4281_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_4282_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_4283_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4284_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_4285_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_4286_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_4287_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_4288_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4289_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4290_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4291_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4292_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4293_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4294_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4295_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_4296_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_4297_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_4298_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_4299_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_4300_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_4301_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(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_4302_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_4303_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4304_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4305_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4306_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4307_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4308_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4309_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4310_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4311_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_4312_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_4313_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_4314_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_4315_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4316_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_4317_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_4318_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_4319_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4320_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4321_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4322_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4323_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4324_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4325_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4326_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_4327_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_4328_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_4329_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_4330_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4331_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_4332_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_4333_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4334_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_4335_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_4336_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_4337_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_4338_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_4339_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_4340_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_4341_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_4342_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_4343_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_4344_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_4345_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_4346_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_4347_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_4348_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_4349_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_4350_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_4351_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_4352_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_4353_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_4354_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_4355_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_4356_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_4357_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_4358_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_4359_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_4360_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_4361_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_4362_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_4363_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_4364_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_4365_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_4366_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_4367_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_4368_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_4369_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_4370_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_4371_abs__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(N) ).

% abs_numeral
tff(fact_4372_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,N) ).

% abs_numeral
tff(fact_4373_abs__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_numeral
tff(fact_4374_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_4375_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_4376_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_4377_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_4378_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_4379_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_4380_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_4381_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_4382_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_4383_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_4384_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_4385_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_4386_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_4387_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_4388_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_4389_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_4390_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_4391_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_4392_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_4393_of__int__abs,axiom,
    ! [X: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,X)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).

% of_int_abs
tff(fact_4394_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_4395_of__int__abs,axiom,
    ! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,X)) ).

% of_int_abs
tff(fact_4396_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_4397_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_4398_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_4399_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_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_4400_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_4401_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_4402_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_4403_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_4404_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_4405_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_4406_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_4407_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(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_4408_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_4409_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_4410_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_4411_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_4412_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_4413_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_4414_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_4415_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_4416_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(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_4417_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_4418_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_4419_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_4420_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_4421_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_4422_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_4423_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_4424_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_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_4425_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_4426_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_4427_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_4428_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_4429_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_4430_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_4431_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_4432_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4433_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4434_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_4435_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_4436_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_4437_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4438_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4439_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_4440_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_4441_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_4442_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4443_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4444_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_4445_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_4446_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4447_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4448_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_4449_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,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4450_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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(aa_int_fun_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_4451_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_4452_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% add_neg_numeral_simps(3)
tff(fact_4453_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N))) ).

% add_neg_numeral_simps(3)
tff(fact_4454_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_4455_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_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_4456_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_4457_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_4458_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_4459_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_4460_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_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_4461_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_4462_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_4463_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_4464_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_4465_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_4466_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4467_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_4468_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_4469_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_4470_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_4471_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_4472_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_4473_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_4474_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_4475_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_4476_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_4477_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_4478_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_4479_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_4480_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_4481_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_4482_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_4483_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_4484_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_4485_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_4486_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_4487_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_4488_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_4489_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_4490_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,N) ).

% abs_neg_numeral
tff(fact_4491_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_4492_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(N) ).

% abs_neg_numeral
tff(fact_4493_abs__neg__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_neg_numeral
tff(fact_4494_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_4495_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_4496_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_4497_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_4498_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_4499_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_4500_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_4501_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_4502_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_4503_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_4504_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% abs_power_minus
tff(fact_4505_abs__power__minus,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),N)) ).

% abs_power_minus
tff(fact_4506_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_4507_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_4508_bit_Odisj__one__left,axiom,
    ! [X: code_integer] : bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_left
tff(fact_4509_bit_Odisj__one__left,axiom,
    ! [X: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_left
tff(fact_4510_bit_Odisj__one__right,axiom,
    ! [X: code_integer] : bit_se1080825931792720795nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_right
tff(fact_4511_bit_Odisj__one__right,axiom,
    ! [X: int] : aa_int_int(bit_se1409905431419307370or_int(X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_right
tff(fact_4512_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_4513_norm__of__nat,axiom,
    ! [N: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% norm_of_nat
tff(fact_4514_norm__of__nat,axiom,
    ! [N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(semiri8010041392384452111omplex,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% norm_of_nat
tff(fact_4515_floor__uminus__of__int,axiom,
    ! [Z: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(uminus_uminus_int,Z) ).

% floor_uminus_of_int
tff(fact_4516_floor__uminus__of__int,axiom,
    ! [Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(uminus_uminus_int,Z) ).

% floor_uminus_of_int
tff(fact_4517_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_4518_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_4519_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_4520_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_4521_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_4522_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_4523_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_4524_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_4525_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_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_4526_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_4527_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4528_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4529_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_4530_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_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_4531_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_4532_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4533_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4534_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_4535_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_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_4536_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_4537_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_4538_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4539_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4540_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_4541_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_4542_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4543_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4544_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4545_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_4546_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_4547_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4548_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4549_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_4550_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(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_4551_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_4552_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),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(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_4553_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_4554_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_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_4555_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_4556_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4557_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4558_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_4559_minus__one__mult__self,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_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_4560_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_4561_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4562_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4563_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_4564_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) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_4565_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_4566_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_4567_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_4568_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_4569_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_4570_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_4571_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_4572_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_4573_floor__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_4574_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_4575_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,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_4576_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_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_4577_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_4578_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_4579_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_4580_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,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4581_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_4582_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_4583_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4584_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_4585_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4586_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_4587_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_4588_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4589_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_4590_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_4591_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_4592_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,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4593_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_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_4594_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_4595_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4596_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_4597_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4598_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_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_4599_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_4600_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4601_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_4602_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,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4603_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_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_4604_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_4605_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4606_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_4607_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4608_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_4609_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_4610_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4611_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_4612_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,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4613_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_4614_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_4615_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4616_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_4617_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,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4618_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_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_4619_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_4620_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4621_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_4622_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_4623_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,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_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_4624_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_4625_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_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_4626_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_4627_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,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_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_4628_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_4629_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_4630_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_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_4631_round__neg__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_4632_round__neg__numeral,axiom,
    ! [N: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_4633_norm__of__int,axiom,
    ! [Z: int] : aa_real_real(real_V7735802525324610683m_real,aa_int_real(ring_1_of_int_real,Z)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Z)) ).

% norm_of_int
tff(fact_4634_norm__of__int,axiom,
    ! [Z: int] : aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(ring_17405671764205052669omplex,Z)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Z)) ).

% norm_of_int
tff(fact_4635_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,aa_num_real(numeral_numeral_real,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4636_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4637_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4638_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_4639_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,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4640_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_4641_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4642_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4643_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,aa_num_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,aa_num_real(numeral_numeral_real,W))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4644_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),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4645_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,aa_num_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,aa_num_real(numeral_numeral_real,W)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4646_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,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4647_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,aa_num_real(numeral_numeral_real,W))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_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,aa_num_real(numeral_numeral_real,W))) ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4648_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_4649_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4650_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,aa_num_real(numeral_numeral_real,W))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4651_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_4652_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4653_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,aa_num_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,aa_num_real(numeral_numeral_real,W)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4654_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,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4655_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,aa_num_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,aa_num_real(numeral_numeral_real,W))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4656_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),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4657_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,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4658_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4659_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4660_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4661_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4662_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_4663_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_4664_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(aa_rat_rat(abs_abs_rat,A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4665_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_4666_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_4667_abs__power2,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_4668_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_4669_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,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_4670_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_4671_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_4672_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_4673_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,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_4674_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_4675_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,aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_4676_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_4677_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)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_4678_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_4679_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)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_4680_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) ).

% of_real_neg_numeral
tff(fact_4681_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_4682_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4683_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4684_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(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4685_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4686_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4687_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)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4688_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4689_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4690_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4691_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4692_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4693_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4694_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(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4695_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4696_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4697_minus__1__div__2__eq,axiom,
    aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_4698_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_4699_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,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4700_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,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4701_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,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4702_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4703_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4704_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,aa_num_num(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_4705_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,aa_num_num(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_4706_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,aa_num_num(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_4707_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_minus_odd
tff(fact_4708_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_minus_odd
tff(fact_4709_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,aa_num_num(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_4710_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,aa_num_num(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_4711_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,aa_num_num(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_4712_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4713_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4714_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,aa_num_num(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_4715_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,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_rat(power_power_rat(aa_rat_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_4716_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,aa_num_num(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_4717_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,aa_num_num(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_4718_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4719_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_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_4720_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_4721_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4722_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_4723_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,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4724_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_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_4725_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_4726_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4727_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_4728_neg__numeral__le__floor,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))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_4729_neg__numeral__le__floor,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(archim6058952711729229775r_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_le_floor
tff(fact_4730_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim3151403230148437115or_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_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% floor_less_neg_numeral
tff(fact_4731_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)))) ) ).

% floor_less_neg_numeral
tff(fact_4732_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,aa_num_real(numeral_numeral_real,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_4733_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_4734_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_4735_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ) ).

% ceiling_less_zero
tff(fact_4736_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_4737_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)) ) ).

% zero_le_ceiling
tff(fact_4738_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,aa_num_real(numeral_numeral_real,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_4739_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))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_4740_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,aa_num_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_4741_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_4742_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_4743_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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_4744_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_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_4745_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,aa_num_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_4746_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_4747_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_4748_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(aa_Cod1783974652533396728nteger(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_4749_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(aa_rat_rat(uminus_uminus_rat,aa_num_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_4750_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,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs
tff(fact_4751_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4752_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,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4753_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4754_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4755_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4756_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,aa_num_num(bit0,one))),N)) = one_one_real ).

% power_minus1_even
tff(fact_4757_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,aa_num_num(bit0,one))),N)) = one_one_int ).

% power_minus1_even
tff(fact_4758_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,aa_num_num(bit0,one))),N)) = one_one_complex ).

% power_minus1_even
tff(fact_4759_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_4760_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_4761_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,aa_num_num(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_4762_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,aa_num_num(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_4763_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,aa_num_num(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_4764_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_4765_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_4766_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,aa_num_num(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_4767_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,aa_num_num(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_4768_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,aa_num_num(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_4769_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_4770_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_4771_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)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_4772_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),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_4773_neg__numeral__less__floor,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))),aa_rat_int(archim3151403230148437115or_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_less_floor
tff(fact_4774_neg__numeral__less__floor,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(archim6058952711729229775r_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(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_less_floor
tff(fact_4775_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_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_rat,X),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% floor_le_neg_numeral
tff(fact_4776_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% floor_le_neg_numeral
tff(fact_4777_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,aa_num_real(numeral_numeral_real,V))),one_one_real))) ) ).

% ceiling_less_neg_numeral
tff(fact_4778_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_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),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat))) ) ).

% ceiling_less_neg_numeral
tff(fact_4779_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,aa_num_real(numeral_numeral_real,V))),one_one_real)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_4780_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))),aa_rat_int(archim2889992004027027881ng_rat,X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_4781_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,aa_num_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_4782_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(aa_Cod1783974652533396728nteger(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_4783_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(aa_rat_rat(uminus_uminus_rat,aa_num_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_4784_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_4785_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,aa_num_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_4786_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(aa_Cod1783974652533396728nteger(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_4787_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(aa_rat_rat(uminus_uminus_rat,aa_num_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_4788_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_4789_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,aa_num_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_4790_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_4791_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(aa_Cod1783974652533396728nteger(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_4792_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(aa_rat_rat(uminus_uminus_rat,aa_num_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_4793_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,aa_num_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_4794_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_4795_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(aa_Cod1783974652533396728nteger(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_4796_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(aa_rat_rat(uminus_uminus_rat,aa_num_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_4797_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,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_4798_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_4799_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_4800_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,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_less_iff
tff(fact_4801_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_less_iff
tff(fact_4802_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_4803_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_4804_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_4805_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_4806_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_4807_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_4808_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_4809_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_4810_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_4811_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_4812_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_4813_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_4814_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_minus_self
tff(fact_4815_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_4816_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_4817_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,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_le_iff
tff(fact_4818_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_le_iff
tff(fact_4819_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_4820_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_4821_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,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% abs_le_D2
tff(fact_4822_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).

% abs_le_D2
tff(fact_4823_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_4824_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_4825_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,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_4826_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,aa_rat_rat(uminus_uminus_rat,A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).

% abs_leI
tff(fact_4827_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_4828_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4829_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4830_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4831_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4832_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_4833_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_4834_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_4835_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_4836_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_4837_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 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_4838_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_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 = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_4839_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_4840_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_4841_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 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_4842_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_4843_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_4844_abs__if__raw,axiom,
    ! [X5: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X5) = aa_real_real(uminus_uminus_real,X5) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_4845_abs__if__raw,axiom,
    ! [X5: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X5) = aa_int_int(uminus_uminus_int,X5) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_4846_abs__if__raw,axiom,
    ! [X5: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X5) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X5) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_4847_abs__if__raw,axiom,
    ! [X5: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X5) = aa_rat_rat(uminus_uminus_rat,X5) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_4848_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_4849_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_4850_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) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_4851_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_4852_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_4853_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_4854_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(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_4855_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_if
tff(fact_4856_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_4857_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_4858_abs__ge__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A))) ).

% abs_ge_self
tff(fact_4859_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_self
tff(fact_4860_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_4861_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_4862_abs__le__D1,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)) ) ).

% abs_le_D1
tff(fact_4863_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% abs_le_D1
tff(fact_4864_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_4865_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_4866_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_4867_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_4868_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_4869_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_4870_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_4871_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_4872_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_4873_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_4874_power__abs,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),N)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N) ).

% power_abs
tff(fact_4875_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_4876_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_4877_ceiling__def,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X))) ).

% ceiling_def
tff(fact_4878_ceiling__def,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(uminus_uminus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,X))) ).

% ceiling_def
tff(fact_4879_floor__minus,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim2889992004027027881ng_rat,X)) ).

% floor_minus
tff(fact_4880_floor__minus,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,X)) = aa_int_int(uminus_uminus_int,aa_real_int(archim7802044766580827645g_real,X)) ).

% floor_minus
tff(fact_4881_ceiling__minus,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_int_int(uminus_uminus_int,aa_rat_int(archim3151403230148437115or_rat,X)) ).

% ceiling_minus
tff(fact_4882_ceiling__minus,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,X)) = aa_int_int(uminus_uminus_int,aa_real_int(archim6058952711729229775r_real,X)) ).

% ceiling_minus
tff(fact_4883_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_4884_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,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_imp_neg_le
tff(fact_4885_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,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_imp_neg_le
tff(fact_4886_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_4887_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_4888_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_le_iff
tff(fact_4889_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_le_iff
tff(fact_4890_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_4891_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_4892_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_minus_iff
tff(fact_4893_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_minus_iff
tff(fact_4894_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_4895_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_4896_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_4897_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,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4898_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,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_4899_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_4900_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_4901_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% less_minus_iff
tff(fact_4902_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% less_minus_iff
tff(fact_4903_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_4904_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_4905_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_less_iff
tff(fact_4906_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_less_iff
tff(fact_4907_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,N) ).

% neg_numeral_neq_numeral
tff(fact_4908_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_4909_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N) ).

% neg_numeral_neq_numeral
tff(fact_4910_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(N) ).

% neg_numeral_neq_numeral
tff(fact_4911_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,N) ).

% neg_numeral_neq_numeral
tff(fact_4912_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% numeral_neq_neg_numeral
tff(fact_4913_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_4914_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% numeral_neq_neg_numeral
tff(fact_4915_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6620942414471956472nteger(M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% numeral_neq_neg_numeral
tff(fact_4916_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_neq_neg_numeral
tff(fact_4917_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_4918_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_4919_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_4920_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_4921_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_4922_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A3) = 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_4923_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4924_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A: complex] :
      ( ( A3 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_4925_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4926_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4927_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_4928_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_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_4929_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_4930_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_4931_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_4932_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_4933_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_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_4934_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_4935_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_4936_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_4937_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_4938_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_4939_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_4940_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_4941_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_4942_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_4943_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,A),B)) ).

% minus_diff_minus
tff(fact_4944_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_4945_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_4946_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_4947_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_4948_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_4949_norm__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(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% norm_triangle_ineq3
tff(fact_4950_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : 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_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).

% norm_triangle_ineq3
tff(fact_4951_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,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_4952_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_4953_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_4954_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))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_4955_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))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_4956_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_4957_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_4958_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_zero
tff(fact_4959_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_4960_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_4961_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_4962_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_4963_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_4964_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_4965_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_4966_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_4967_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_4968_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq
tff(fact_4969_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_4970_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq
tff(fact_4971_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(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_4972_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D3: 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)),D3))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3))) ) ) ).

% abs_mult_less
tff(fact_4973_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D3: 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)),D3))
       => 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),D3))) ) ) ).

% abs_mult_less
tff(fact_4974_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D3))) ) ) ).

% abs_mult_less
tff(fact_4975_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D3: 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)),D3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D3))) ) ) ).

% abs_mult_less
tff(fact_4976_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)))) ).

% abs_triangle_ineq2_sym
tff(fact_4977_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_4978_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_4979_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(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_4980_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq3
tff(fact_4981_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_4982_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq3
tff(fact_4983_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(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq3
tff(fact_4984_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq2
tff(fact_4985_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_4986_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq2
tff(fact_4987_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq2
tff(fact_4988_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_4989_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_4990_norm__of__real__diff,axiom,
    ! [B: real,A: 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(real_V1803761363581548252l_real,B)),aa_real_real(real_V1803761363581548252l_real,A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).

% norm_of_real_diff
tff(fact_4991_norm__of__real__diff,axiom,
    ! [B: real,A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,B)),aa_real_complex(real_V4546457046886955230omplex,A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).

% norm_of_real_diff
tff(fact_4992_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_4993_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_4994_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_4995_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_4996_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_4997_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_4998_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_4999_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5000_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_5001_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_5002_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,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_5003_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_le_numeral
tff(fact_5004_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_5005_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_5006_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% zero_neq_neg_numeral
tff(fact_5007_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_5008_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_5009_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% zero_neq_neg_numeral
tff(fact_5010_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% zero_neq_neg_numeral
tff(fact_5011_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,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_5012_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_5013_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_less_numeral
tff(fact_5014_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_5015_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5016_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_5017_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5018_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_5019_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_5020_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_5021_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_5022_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_5023_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_5024_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_5025_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_5026_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_5027_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5028_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5029_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5030_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5031_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5032_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_5033_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5034_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_5035_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5036_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5037_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_5038_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5039_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_5040_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5041_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5042_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_5043_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_5044_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_5045_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_5046_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5047_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_5048_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_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_5049_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_5050_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_5051_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5052_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_5053_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_5054_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_5055_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5056_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5057_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_5058_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_5059_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_5060_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_5061_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_5062_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_5063_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_5064_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_5065_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_5066_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_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_5067_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_5068_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_5069_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(times_times_rat(X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_5070_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% one_neq_neg_numeral
tff(fact_5071_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_5072_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_5073_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% one_neq_neg_numeral
tff(fact_5074_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% one_neq_neg_numeral
tff(fact_5075_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5076_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_5077_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5078_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6620942414471956472nteger(N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5079_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5080_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_5081_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_5082_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5083_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_5084_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_5085_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5086_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_5087_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_5088_group__cancel_Osub2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B4) = 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_5089_group__cancel_Osub2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5090_group__cancel_Osub2,axiom,
    ! [B4: complex,K: complex,B: complex,A: complex] :
      ( ( B4 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5091_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B4 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5092_group__cancel_Osub2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5093_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_5094_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_5095_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_5096_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_5097_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_5098_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_5099_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5100_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_5101_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5102_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5103_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_5104_of__int__of__nat,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,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) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_5105_of__int__of__nat,axiom,
    ! [K: 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_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,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_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_5106_of__int__of__nat,axiom,
    ! [K: 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_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,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_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_5107_of__int__of__nat,axiom,
    ! [K: 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) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,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) = aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_5108_of__int__of__nat,axiom,
    ! [K: 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_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,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_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K)) ) ) ) ).

% of_int_of_nat
tff(fact_5109_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5110_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5111_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5112_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_5113_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(aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5114_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(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5115_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_5116_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_5117_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)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_5118_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(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_5119_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_5120_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% abs_div_pos
tff(fact_5121_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_5122_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_5123_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(aa_rat_rat(abs_abs_rat,A)),N))) ).

% zero_le_power_abs
tff(fact_5124_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_5125_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5126_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_5127_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5128_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(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5129_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D3)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D3))))) ).

% abs_diff_triangle_ineq
tff(fact_5130_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3)))),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),D3))))) ).

% abs_diff_triangle_ineq
tff(fact_5131_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D3)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D3))))) ).

% abs_diff_triangle_ineq
tff(fact_5132_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D3)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D3))))) ).

% abs_diff_triangle_ineq
tff(fact_5133_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq4
tff(fact_5134_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_5135_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq4
tff(fact_5136_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(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_5137_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_5138_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_5139_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5140_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(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5141_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_5142_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_5143_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_5144_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_5145_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,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5146_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_le_neg_numeral
tff(fact_5147_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_le_neg_numeral
tff(fact_5148_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_5149_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_5150_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_5151_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_5152_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_5153_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_5154_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_5155_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_5156_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_5157_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,aa_num_real(numeral_numeral_real,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5158_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_5159_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).

% not_zero_less_neg_numeral
tff(fact_5160_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_zero_less_neg_numeral
tff(fact_5161_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_5162_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_5163_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_5164_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_5165_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_5166_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_5167_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_5168_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_5169_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_5170_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_5171_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_5172_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_5173_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_5174_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_5175_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_5176_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_5177_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,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_le_neg_numeral
tff(fact_5178_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_le_neg_numeral
tff(fact_5179_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_le_neg_numeral
tff(fact_5180_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_5181_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_5182_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_5183_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_5184_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_5185_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,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_5186_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_5187_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_5188_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_5189_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)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_le_numeral
tff(fact_5190_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_le_numeral
tff(fact_5191_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_le_numeral
tff(fact_5192_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_5193_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_5194_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_5195_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_5196_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_5197_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_5198_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_5199_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_5200_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_5201_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)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_less_numeral
tff(fact_5202_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_5203_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_less_numeral
tff(fact_5204_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_less_numeral
tff(fact_5205_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_5206_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_5207_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_5208_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_5209_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,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_less_neg_numeral
tff(fact_5210_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_5211_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_less_neg_numeral
tff(fact_5212_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_less_neg_numeral
tff(fact_5213_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,aa_num_real(numeral_numeral_real,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5214_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_5215_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5216_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_5217_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_5218_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_5219_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5220_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_5221_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_5222_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5223_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_5224_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_5225_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_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_5226_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_5227_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_5228_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_5229_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: 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))),E2))
     => 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))),E2)) ) ).

% norm_triangle_lt
tff(fact_5230_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: 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))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_5231_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_5232_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_5233_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_5234_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_5235_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5236_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,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_5237_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_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_5238_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_5239_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_5240_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_5241_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_5242_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_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_5243_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_5244_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_5245_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_5246_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_5247_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_5248_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_5249_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_5250_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_5251_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_5252_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_5253_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: 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))),E2))
     => 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))),E2)) ) ).

% norm_triangle_le
tff(fact_5254_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: 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))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_5255_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_5256_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_5257_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_5258_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_5259_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_5260_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_5261_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_5262_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_5263_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_5264_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_5265_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_5266_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_5267_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_5268_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_5269_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E2: 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))),E2))
     => 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))),E2)) ) ).

% norm_triangle_le_diff
tff(fact_5270_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E2: 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))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E2)) ) ).

% norm_triangle_le_diff
tff(fact_5271_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_5272_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_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_5273_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_5274_power__minus,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_minus
tff(fact_5275_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_5276_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_5277_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_5278_norm__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(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% norm_triangle_ineq2
tff(fact_5279_norm__triangle__ineq2,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(minus_minus_complex(A),B)))) ).

% norm_triangle_ineq2
tff(fact_5280_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,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5281_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,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5282_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,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5283_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5284_power__minus__Bit0,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5285_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [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))),D6))
               => ( 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_5286_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_5287_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_5288_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_5289_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_5290_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_5291_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_5292_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_5293_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_5294_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).

% abs_add_one_gt_zero
tff(fact_5295_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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_5296_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,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_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).

% of_int_leD
tff(fact_5297_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_5298_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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_5299_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_5300_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_5301_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_5302_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_5303_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_5304_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_5305_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_5306_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D3: 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),D3)))),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),D3))))) ).

% norm_diff_triangle_ineq
tff(fact_5307_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D3: 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),D3)))),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),D3))))) ).

% norm_diff_triangle_ineq
tff(fact_5308_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_5309_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_5310_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_5311_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(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,aa_rat_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_5312_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_5313_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_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,aa_rat_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)),aa_rat_rat(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_5314_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_5315_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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5316_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_5317_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,aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5318_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_5319_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),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5320_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_5321_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,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5322_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,aa_num_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,aa_num_real(numeral_numeral_real,W))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5323_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_5324_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5325_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_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,aa_num_real(numeral_numeral_real,W))),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5326_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_5327_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5328_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_5329_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_5330_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5331_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_5332_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_5333_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_5334_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_5335_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_5336_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5337_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_5338_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_5339_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_5340_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_5341_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_5342_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_5343_even__minus,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),A)) ) ).

% even_minus
tff(fact_5344_even__minus,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).

% even_minus
tff(fact_5345_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,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5346_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5347_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5348_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5349_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5350_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [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))),D6))
               => ( 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_5351_round__diff__minimal,axiom,
    ! [Z: real,M: int] : 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,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M))))) ).

% round_diff_minimal
tff(fact_5352_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,M))))) ).

% round_diff_minimal
tff(fact_5353_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,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5354_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5355_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(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,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5356_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,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5357_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_5358_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_5359_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_5360_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,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,X) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_5361_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,aa_num_num(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_5362_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,aa_num_num(bit0,one))),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_even_abs
tff(fact_5363_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,aa_num_num(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_5364_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,aa_num_num(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_5365_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_5366_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),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(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,aa_rat_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_5367_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_5368_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_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,aa_rat_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)),aa_rat_rat(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_5369_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_5370_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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5371_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_5372_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,aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5373_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_5374_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),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5375_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_5376_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,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5377_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,aa_num_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,aa_num_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,aa_num_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,aa_num_real(numeral_numeral_real,W)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5378_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,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5379_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,aa_num_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,aa_num_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,aa_num_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,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5380_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,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5381_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_5382_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_5383_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,aa_num_num(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_5384_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(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_5385_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_5386_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_5387_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_5388_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,aa_num_num(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,aa_num_num(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_5389_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,aa_num_num(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,aa_num_num(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_5390_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,aa_num_num(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,aa_num_num(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_5391_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,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5392_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,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_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,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_5393_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_5394_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_5395_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_5396_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(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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_5397_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(aa_rat_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(aa_rat_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_5398_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,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_5399_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_5400_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_5401_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_5402_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,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_5403_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_5404_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,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5405_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,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_5406_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,aa_num_num(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_5407_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,aa_num_num(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_5408_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,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_5409_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,aa_num_num(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_5410_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,aa_num_num(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_5411_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,aa_num_num(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_5412_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,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_5413_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,aa_num_num(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_5414_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,aa_num_num(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_5415_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,aa_num_num(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_5416_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,aa_num_num(bit0,one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(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_5417_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,aa_num_num(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_5418_square__norm__one,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_5419_square__norm__one,axiom,
    ! [X: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_5420_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,aa_num_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,aa_num_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,aa_num_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,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5421_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,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5422_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,aa_num_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,aa_num_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,aa_num_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,aa_num_real(numeral_numeral_real,W)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5423_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,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5424_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,aa_num_num(bit0,one)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_5425_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,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_5426_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_5427_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_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,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_5428_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,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_5429_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5430_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,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5431_minus__power__mult__self,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5432_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,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5433_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5434_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)),aa_nat_rat(power_power_rat(aa_rat_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,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_5435_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,aa_num_num(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,aa_num_num(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_5436_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,aa_num_num(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,aa_num_num(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_5437_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,aa_num_num(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,aa_num_num(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_5438_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,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% minus_one_power_iff
tff(fact_5439_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,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_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,aa_num_num(bit0,one))),N))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% minus_one_power_iff
tff(fact_5440_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_5441_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,aa_num_num(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,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_5442_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_5443_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs1
tff(fact_5444_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs2
tff(fact_5445_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_5446_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,aa_num_num(bit0,one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5447_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,aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5448_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,aa_num_num(bit0,one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5449_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5450_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_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,aa_num_num(bit0,one))),N))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_5451_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(suc,N)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_5452_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_5453_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K)),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_5454_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_5455_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,aa_num_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_5456_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_5457_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_5458_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = N ) ) ).

% round_unique'
tff(fact_5459_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,N)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N ) ) ).

% round_unique'
tff(fact_5460_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real)) ).

% cos_x_y_le_one
tff(fact_5461_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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_5462_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,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_5463_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_5464_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_5465_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(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_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_5466_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X4: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5467_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X4),aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5468_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5469_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5470__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [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,aa_num_num(bit0,one))),m)))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I2 )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [X5: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),na) = I2 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),X5))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),ma)) ) ) ) ) ) ).

% "5.hyps"(9)
tff(fact_5471_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_5472_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,aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_5473_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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_5474_pochhammer__double,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_5475_pochhammer__double,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_5476_pochhammer__double,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_5477_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_5478_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_5479_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_5480_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_5481_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_5482_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_5483_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_5484_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_5485_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_5486_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_5487_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_5488_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_5489_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_5490_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_5491_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_5492_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_5493_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_5494_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_5495_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_5496_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_5497_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_5498_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_5499_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_5500_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_5501_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_5502_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_5503_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_5504_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_5505_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_5506_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_5507_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_5508_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_5509_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_5510_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_5511_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_5512_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_5513_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_5514_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_5515_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_5516_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_5517_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_5518_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_5519_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5520_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_5521_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_5522_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_5523_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_5524_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_5525_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_5526__C5_Ohyps_C_I5_J,axiom,
    ! [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,aa_num_num(bit0,one))),m)))
     => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I2)),X_12))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I2)) ) ) ).

% "5.hyps"(5)
tff(fact_5527_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_5528_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5529_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_5530_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_5531_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_5532_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_5533_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,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_5534_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_5535_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_5536_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,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_5537_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_5538_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_5539_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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).

% even_mod_2_iff
tff(fact_5540_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,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_mod_2_iff
tff(fact_5541_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).

% even_mod_2_iff
tff(fact_5542_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,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_5543_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_5544_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_5545_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_5546_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,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5547_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5548_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5549_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,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5550_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5551_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5552_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,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_5553_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5554_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,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_5555_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5556_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,aa_num_num(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,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_5557_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,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_5558_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_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_5559_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5560_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5561_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5562_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,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_5563_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,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_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_5564_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,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_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_5565__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) ) ).

% "5.hyps"(6)
tff(fact_5566_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),aa_num_real(numeral_numeral_real,B)))) = aa_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_5567_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5568_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,aa_num_num(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(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5569_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(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(aa_num_num(bit0,one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5570_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_int_real(ring_1_of_int_real,X)),N) = aa_int_real(ring_1_of_int_real,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_5571_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_int_complex(ring_17405671764205052669omplex,X)),N) = aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_5572_pochhammer__of__int,axiom,
    ! [X: int,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_int_rat(ring_1_of_int_rat,X)),N) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(comm_s4660882817536571857er_int(X),N)) ).

% pochhammer_of_int
tff(fact_5573_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_nat_complex(semiri8010041392384452111omplex,X)),N) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_5574_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_nat_real(semiri5074537144036343181t_real,X)),N) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_5575_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_5576_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_5577_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_nat_int(semiri1314217659103216013at_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N)) ).

% pochhammer_of_nat
tff(fact_5578_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_5579_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_5580_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_5581_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_5582_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_5583_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_5584_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_5585_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_5586_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_5587_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_5588_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_5589_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_5590_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% arctan_le_iff
tff(fact_5591_arctan__monotone_H,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(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone'
tff(fact_5592_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_5593_uminus__dvd__conv_I2_J,axiom,
    ! [D3: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(uminus_uminus_int,T))) ) ).

% uminus_dvd_conv(2)
tff(fact_5594_uminus__dvd__conv_I1_J,axiom,
    ! [D3: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,D3)),T)) ) ).

% uminus_dvd_conv(1)
tff(fact_5595_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_5596_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_5597_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_5598_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_5599_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_5600_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_5601_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_5602_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5603_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5604_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5605_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_5606_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5607_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_5608_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_5609_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_5610_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_5611_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_5612_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_5613_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_5614_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_5615_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_5616_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_5617_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_5618_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_5619_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_5620_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_5621_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_5622_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_5623_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M4),zero_zero_nat))
     => ( ! [M4: 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,M4),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M4),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_5624_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_5625_mod__eq__0D,axiom,
    ! [M: nat,D3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D3) = zero_zero_nat )
     => ? [Q4: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D3),Q4) ) ).

% mod_eq_0D
tff(fact_5626_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_5627_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_5628_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_5629_pochhammer__pos,axiom,
    ! [X: real,N: 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,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_pos
tff(fact_5630_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( 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),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_pos
tff(fact_5631_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_pos
tff(fact_5632_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( 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,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_pos
tff(fact_5633_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_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_5634_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) != 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(comm_s7457072308508201937r_real,A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5635_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5636_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = 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(comm_s7457072308508201937r_real,A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5637_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5638_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_5639_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_5640_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_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_5641_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_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_5642_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_5643_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_5644_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_5645_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_5646_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_5647_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_5648_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_5649_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_5650_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = 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,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5651_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5652_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5653_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_5654_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_5655_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_5656_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: 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,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_5657_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_5658_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_5659_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_5660_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_5661_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_5662_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_5663_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_5664_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_5665_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_5666_div__less__mono,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B4))
     => ( 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,A3),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),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,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),N))) ) ) ) ) ).

% div_less_mono
tff(fact_5667_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_5668_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( 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,Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_5669_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
     => ( 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,Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_5670_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))
       => ? [Q4: 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),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_5671_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] : A3 = 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,A3),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N)) ).

% div_mod_decomp
tff(fact_5672_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( 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_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_nonneg
tff(fact_5673_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( 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,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).

% pochhammer_nonneg
tff(fact_5674_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).

% pochhammer_nonneg
tff(fact_5675_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( 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),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).

% pochhammer_nonneg
tff(fact_5676_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: 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),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q3) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5677_dvd__imp__le__int,axiom,
    ! [I: int,D3: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D3)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_5678_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_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_5679_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_5680_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_5681_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_5682_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_5683_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_5684_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_5685_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_5686_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_5687_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_5688_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_5689_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_5690_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_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(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_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_5691_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,aa_num_num(bit0,one))),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_5692_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_5693_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_5694_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)) = aa_rat_rat(divide_divide_rat(aa_rat_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_5695_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 )
         => ! [I4: 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),I4)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_5696_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_5697_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(aa_int_fun_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_5698_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(aa_int_fun_int_int(times_times_int,M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_5699_real__of__nat__div__aux,axiom,
    ! [X: nat,D3: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D3)) = 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),D3))),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),D3))),aa_nat_real(semiri5074537144036343181t_real,D3))) ).

% real_of_nat_div_aux
tff(fact_5700_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_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_5701_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N)) ).

% pochhammer_rec'
tff(fact_5702_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)) ).

% pochhammer_rec'
tff(fact_5703_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_5704_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_5705_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).

% pochhammer_rec'
tff(fact_5706_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),N)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))) ).

% pochhammer_Suc
tff(fact_5707_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_5708_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_5709_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_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(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_5710_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_5711_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5712_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5713_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5714_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5715_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5716_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5717_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5718_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5719_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5720_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5721_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5722_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5723_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5724_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5725_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5726_pochhammer__of__nat__eq__0__lemma_H,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(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5727_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5728_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5729_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),N)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),M)) ).

% pochhammer_product'
tff(fact_5730_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_5731_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).

% pochhammer_product'
tff(fact_5732_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_5733_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_5734_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_5735_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_5736_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_5737_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_5738_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B4))
     => ( 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(divide_divide_int(B4),N)),aa_int_int(divide_divide_int(A3),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_5739_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(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_5740_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(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_5741_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_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(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_5742_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5743_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_5744_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(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_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_5745_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,aa_num_num(bit0,one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5746_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,aa_num_num(bit0,one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5747_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5748_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,aa_num_num(bit0,one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5749_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,aa_num_num(bit0,one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5750_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5751_take__bit__eq__mod,axiom,
    ! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(N,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% take_bit_eq_mod
tff(fact_5752_take__bit__eq__mod,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_eq_mod
tff(fact_5753_take__bit__eq__mod,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = 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,aa_num_num(bit0,one))),N)) ).

% take_bit_eq_mod
tff(fact_5754_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_5755_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).

% even_abs_add_iff
tff(fact_5756_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).

% even_add_abs_iff
tff(fact_5757_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(aa_int_fun_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(aa_int_fun_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_5758_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,aa_num_num(bit0,one))),N)) ).

% take_bit_nat_def
tff(fact_5759_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_5760_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),N) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_5761_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_5762_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_5763_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_5764_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_5765_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: 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(aa_int_fun_int_int(times_times_int,Z),Z4)) = 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,Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_5766_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,aa_num_num(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,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5767_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5768_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5769_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,aa_num_num(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,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_5770_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_5771_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_5772_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_5773_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_5774_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_5775_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_5776_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_5777_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_5778_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,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5779_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5780_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5781_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B4))
     => ( 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,A3),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,B4),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),N))) ) ) ).

% verit_le_mono_div
tff(fact_5782_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(aa_int_fun_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_5783_decr__lemma,axiom,
    ! [D3: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D3))),Z)) ) ).

% decr_lemma
tff(fact_5784_incr__lemma,axiom,
    ! [D3: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D3)))) ) ).

% incr_lemma
tff(fact_5785_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_5786_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_5787_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_5788_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_5789_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_5790_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,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_5791_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa_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,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_5792_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(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5793_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,aa_num_num(bit0,one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5794_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5795_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5796_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,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5797_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(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5798_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5799_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y)) ) ).

% compl_le_swap2
tff(fact_5800_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X)))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).

% compl_le_swap1
tff(fact_5801_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).

% compl_mono
tff(fact_5802_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(aa_int_fun_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_5803_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_5804_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_5805_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_5806_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = 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)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_5807_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_5808_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_5809_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_5810_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_5811_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = 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)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_5812_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_5813_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_5814_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_5815_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(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_5816_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5817_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,aa_num_num(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,aa_num_num(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_5818_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(modulo_modulo_int(X),M) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5819_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5820_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,aa_num_num(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,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5821_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5822_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5823_take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5824_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5825_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5826_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5827_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5828_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5829_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5830_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5831_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5832_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5833_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5834_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,aa_num_num(bit0,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_5835_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(aa_int_fun_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_5836_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_5837_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_5838_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(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5839_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,aa_num_num(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,aa_num_num(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,aa_num_num(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_5840_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5841_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,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5842_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
      & ( ( N != zero_zero_nat )
       => ( bit_se1745604003318907178nteger(N,A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_5843_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_5844_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),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(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% take_bit_rec
tff(fact_5845_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_5846_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_5847_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_5848_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_5849_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,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_5850_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_5851_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_5852_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_5853_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_5854_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_5855_inthall,axiom,
    ! [Xs: list_option_nat,P: fun_option_nat_bool,N: nat] :
      ( ! [X4: option_nat] :
          ( member_option_nat(X4,set_option_nat2(Xs))
         => pp(aa_option_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6086282163384603972on_nat(Xs)))
       => pp(aa_option_nat_bool(P,nth_option_nat(Xs,N))) ) ) ).

% inthall
tff(fact_5856_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_real2(Xs)))
         => pp(aa_real_bool(P,X4)) )
     => ( 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_5857_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N: nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => pp(aa_set_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),N))) ) ) ).

% inthall
tff(fact_5858_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
     => ( 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_5859_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_bool_bool(P,X4)) )
     => ( 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_5860_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X4)) )
     => ( 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_5861_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P,X4)) )
     => ( 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_5862_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_5863_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_5864_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_5865_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_5866_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_5867_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_5868_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_5869_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_5870_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,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_5871_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,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5872_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5873_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,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5874__C5_Ohyps_C_I1_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "5.hyps"(1)
tff(fact_5875_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_5876_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_5877_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_5878_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_5879_length__pos__if__in__set,axiom,
    ! [X: option_nat,Xs: list_option_nat] :
      ( member_option_nat(X,set_option_nat2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6086282163384603972on_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_5880_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_5881_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_5882_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(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_5883_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_5884_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_5885_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_5886_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_5887_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => pp(aa_bool_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_5888_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_5889_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_int_bool(P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_5890_all__nth__imp__all__set,axiom,
    ! [Xs: list_option_nat,P: fun_option_nat_bool,X: option_nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6086282163384603972on_nat(Xs)))
         => pp(aa_option_nat_bool(P,nth_option_nat(Xs,I3))) )
     => ( member_option_nat(X,set_option_nat2(Xs))
       => pp(aa_option_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_5891_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_5892_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X: set_nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),I3))) )
     => ( member_set_nat(X,set_set_nat2(Xs))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_5893_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))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_5894_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_5895_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_5896_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_5897_in__set__conv__nth,axiom,
    ! [X: option_nat,Xs: list_option_nat] :
      ( member_option_nat(X,set_option_nat2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6086282163384603972on_nat(Xs)))
          & ( nth_option_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5898_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5899_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3254054031482475050et_nat(Xs)))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5900_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5901_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I4))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_5902_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5903_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5904_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)))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X4)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_5905_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)))
     => ( ! [X4: bool] :
            ( member_bool(X4,set_bool2(Xs))
           => pp(aa_bool_bool(P,X4)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_5906_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)))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X4)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_5907_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)))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
           => pp(aa_int_bool(P,X4)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_5908_nth__mem,axiom,
    ! [N: nat,Xs: list_option_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6086282163384603972on_nat(Xs)))
     => member_option_nat(nth_option_nat(Xs,N),set_option_nat2(Xs)) ) ).

% nth_mem
tff(fact_5909_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_5910_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),N),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_5911_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)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_5912_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_5913_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_5914_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_5915_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_5916_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_5917_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_5918_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_5919_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_5920_zmod__eq__0__iff,axiom,
    ! [M: int,D3: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D3) = zero_zero_int )
    <=> ? [Q5: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D3),Q5) ) ).

% zmod_eq_0_iff
tff(fact_5921_zmod__eq__0D,axiom,
    ! [M: int,D3: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D3) = zero_zero_int )
     => ? [Q4: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D3),Q4) ) ).

% zmod_eq_0D
tff(fact_5922_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_5923_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(bit_ri631733984087533419it_int(N),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5924_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) = aa_int_int(if_fun_int_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M),bit_se2923211474154528505it_int(N),bit_ri631733984087533419it_int(M)),A) ).

% signed_take_bit_take_bit
tff(fact_5925_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_5926_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_5927_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_5928_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_5929_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_5930_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(modulo_modulo_int(A3),N) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B4),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A3),N)),aa_int_int(divide_divide_int(B4),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_5931_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(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_5932_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(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_5933_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_5934_div__mod__decomp__int,axiom,
    ! [A3: int,N: int] : A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A3),N)),N)),aa_int_int(modulo_modulo_int(A3),N)) ).

% div_mod_decomp_int
tff(fact_5935_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(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_5936_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(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5937_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_5938_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_5939_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri631733984087533419it_int(N),A)) = aa_int_int(bit_se2923211474154528505it_int(M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_5940_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_5941_real__of__int__div__aux,axiom,
    ! [X: int,D3: 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,D3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(X),D3))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(X),D3))),aa_int_real(ring_1_of_int_real,D3))) ).

% real_of_int_div_aux
tff(fact_5942_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,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_5943_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( 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,aa_num_num(bit0,one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_5944_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( 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,aa_num_num(bit0,one))),K6)),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),K6))) ) ) ).

% binomial_mono
tff(fact_5945_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,aa_num_num(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,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_5946_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))
         => ! [I4: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I4: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_5947_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),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_5948_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),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_5949_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(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5950_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(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5951_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,aa_num_num(bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_5952_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,aa_num_num(bit0,one))),N)) ).

% take_bit_int_def
tff(fact_5953_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(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_5954_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(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_5955_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(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_5956_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(M,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).

% even_signed_take_bit_iff
tff(fact_5957_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,aa_num_num(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,aa_num_num(bit0,one))),A)) ) ).

% even_signed_take_bit_iff
tff(fact_5958_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,aa_num_num(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_5959_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( 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,aa_num_num(bit0,one))),K6)),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),K6))) ) ) ).

% binomial_strict_mono
tff(fact_5960_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( 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,aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_5961_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_5962_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B4))
     => ( 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A3),N)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B4),N)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(divide_divide_int(B4),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_5963_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,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_5964_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,aa_num_num(bit0,one))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5965_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(divide_divide_int(N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I4: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_5966_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(divide_divide_int(N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I4: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_5967_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,aa_num_num(bit0,one))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5968_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,aa_num_num(bit0,one))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5969_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,aa_num_num(bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5970_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_5971_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_5972_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_5973_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_5974_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,aa_num_num(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(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5975_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,aa_num_num(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,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5976_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,aa_num_num(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,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5977_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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_5978_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_5979_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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5980_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,aa_num_num(bit0,one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5981_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5982_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5983_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5984_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I4))
            <=> pp(aa_nat_bool(nth_bool(Ys),I4)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5985_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5986_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5987_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_12: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),nth_VEBT_VEBT(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_5988_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_12: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),aa_nat_bool(nth_bool(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_5989_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(nth_nat(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_5990_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
         => ? [X_12: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),aa_nat_int(nth_int(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_5991_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_5992_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_5993_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_5994_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_5995_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_5996_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_5997_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_5998_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_5999_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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_6000_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_6001_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_6002_binomial__symmetric,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(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_6003_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_6004_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_6005_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_6006_binomial__ge__n__over__k__pow__k,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(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6007_binomial__ge__n__over__k__pow__k,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(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6008_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,aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_6009_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_6010_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_6011_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_6012_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_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,aa_num_num(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,aa_num_num(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_6013_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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,aa_num_num(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_6014_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Deg) )
     => ( pp(aa_nat_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% height_double_log_univ_size
tff(fact_6015_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% heigt_uplog_rel
tff(fact_6016_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,one))),N))) ) ) ).

% helpypredd
tff(fact_6017_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,one))),N))) ) ) ).

% helpyd
tff(fact_6018_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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,aa_num_num(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,aa_num_num(bit0,one))),N))) ) ).

% two_powr_height_bound_deg
tff(fact_6019_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(vEBT_invar_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_6020_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_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,aa_num_num(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,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_6021_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(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_6022_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_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,aa_num_num(bit0,one))),N))) ) ) ).

% member_bound
tff(fact_6023_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,one))),N))) ) ) ).

% misiz
tff(fact_6024_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(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,vEBT_Node(Info,Deg,TreeList,Summary)))) ) ).

% height_compose_child
tff(fact_6025_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_6026_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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 = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2,S3) ) ).

% deg_SUcn_Node
tff(fact_6027_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_6028_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_6029_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% mint_member
tff(fact_6030_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,vEBT_Node(Info,Deg,TreeList,Summary)))) ).

% height_compose_summary
tff(fact_6031_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_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_6032_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_6033_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6034_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6035_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,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_6036_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
            & pp(aa_nat_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,aa_num_num(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,aa_num_num(bit0,one)))))) ) ) ) ) ).

% member_inv
tff(fact_6037_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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))
          & ? [M4: nat] :
              ( ( some_nat(M4) = vEBT_vebt_mint(Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_6038_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,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,aa_num_num(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,aa_num_num(bit0,one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_6039_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_6040_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) )
          & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_6041_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,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_6042_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_6043_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,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(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_6044_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,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(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_6045_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(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [Z5: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z5))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z5)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z5)) ) ) ) ).

% succ_member
tff(fact_6046_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(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [Z5: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z5))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z5),Y)) ) ) ) ).

% pred_member
tff(fact_6047_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( pp(aa_nat_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_6048_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_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_6049_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_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_6050_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_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_6051_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_6052_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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,aa_num_num(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,aa_num_num(bit0,one))),M)))
                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_12))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),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,aa_num_num(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,aa_num_num(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))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),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,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_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_6053_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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,aa_num_num(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,aa_num_num(bit0,one))),M)))
                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_12))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),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,aa_num_num(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,aa_num_num(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))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),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,X4),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_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_6054_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% nested_mint
tff(fact_6055_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: 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(Q3),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,aa_num_num(bit0,one))),Q3)),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(Q3),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,aa_num_num(bit0,one))),Q3)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_6056_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: 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(Q3,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),one_one_int),aa_int_int(aa_int_fun_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(Q3,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3),R2) ) ) ) ).

% divmod_step_eq
tff(fact_6057_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),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(Q3,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),R2) ) ) ) ).

% divmod_step_eq
tff(fact_6058_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,aa_num_num(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,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_pred(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_6059_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,aa_num_num(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,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_succ(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_6060_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)))) ) ).

% height_node
tff(fact_6061_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_6062_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_6063_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y5: nat] : X = some_nat(Y5) ) ).

% not_None_eq
tff(fact_6064_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_6065_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y5: num] : X = some_num(Y5) ) ).

% not_None_eq
tff(fact_6066_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y5: nat] : X != some_nat(Y5)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_6067_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y5)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_6068_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y5: num] : X != some_num(Y5)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_6069_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(aa_option_nat_nat(the_nat,Option)) = Option ) ) ).

% option.collapse
tff(fact_6070_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_6071_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_6072_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(aa_option_nat_nat(the_nat,Option)) ) ) ).

% option.exhaust_sel
tff(fact_6073_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_6074_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_6075_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,V3: nat] : X != produc2004789812759622991on_nat(Uw,produc5098337634421038937on_nat(some_nat(V3),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_bool,X4: nat,Y3: nat] : X != produc2004789812759622991on_nat(F2,produc5098337634421038937on_nat(some_nat(X4),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_6076_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,V3: product_prod_nat_nat] : X != produc5961576194287291153at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V3),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr248683899324676269t_bool,X4: product_prod_nat_nat,Y3: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X4),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_6077_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,V3: num] : X != produc8433281988922931063on_num(Uw,produc8585076106096196333on_num(some_num(V3),none_num))
       => ~ ! [F2: fun_num_fun_num_bool,X4: num,Y3: num] : X != produc8433281988922931063on_num(F2,produc8585076106096196333on_num(some_num(X4),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_6078_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,V3: nat] : X != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(some_nat(V3),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B3: nat] : X != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A4),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_6079_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,V3: product_prod_nat_nat] : X != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V3),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_6080_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,V3: num] : X != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(some_num(V3),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A4: num,B3: num] : X != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A4),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_6081_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( aa_option_nat_nat(the_nat,Option) = aa_option_nat_nat(the_nat,Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_6082_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_6083_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_6084_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_6085_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_6086_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_6087_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_6088_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_6089_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_6090_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] : Y != some_nat(X22) ) ).

% option.exhaust
tff(fact_6091_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_6092_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != some_num(X22) ) ).

% option.exhaust
tff(fact_6093_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_12: option_nat] : pp(aa_option_nat_bool(P,X_12))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_ex
tff(fact_6094_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_12: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_12))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        | ? [X3: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X3))) ) ) ).

% split_option_ex
tff(fact_6095_split__option__ex,axiom,
    ! [P: fun_option_num_bool] :
      ( ? [X_12: option_num] : pp(aa_option_num_bool(P,X_12))
    <=> ( pp(aa_option_num_bool(P,none_num))
        | ? [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_ex
tff(fact_6096_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_12: option_nat] : pp(aa_option_nat_bool(P,X_12))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X3: nat] : pp(aa_option_nat_bool(P,some_nat(X3))) ) ) ).

% split_option_all
tff(fact_6097_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_12: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_12))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        & ! [X3: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X3))) ) ) ).

% split_option_all
tff(fact_6098_split__option__all,axiom,
    ! [P: fun_option_num_bool] :
      ( ! [X_12: option_num] : pp(aa_option_num_bool(P,X_12))
    <=> ( pp(aa_option_num_bool(P,none_num))
        & ! [X3: num] : pp(aa_option_num_bool(P,some_num(X3))) ) ) ).

% split_option_all
tff(fact_6099_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,B3: nat] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_nat(B3) )
               => 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_6100_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,B3: product_prod_nat_nat] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => 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_6101_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,B3: num] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_num(B3) )
               => 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_6102_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,B3: nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_nat(B3) )
               => 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_6103_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,B3: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => 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_6104_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,B3: num] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_num(B3) )
               => 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_6105_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,B3: nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_nat(B3) )
               => 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_6106_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,B3: product_prod_nat_nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => 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_6107_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,B3: num] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_num(B3) )
               => 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_6108_option_Osel,axiom,
    ! [X2: nat] : aa_option_nat_nat(the_nat,some_nat(X2)) = X2 ).

% option.sel
tff(fact_6109_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).

% option.sel
tff(fact_6110_option_Osel,axiom,
    ! [X2: num] : the_num(some_num(X2)) = X2 ).

% option.sel
tff(fact_6111_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_6112_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_6113_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_6114_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
         => pp(aa_nat_bool(P,M3)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less
tff(fact_6115_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N))
          & pp(aa_nat_bool(P,M3)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less
tff(fact_6116_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_6117_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_6118_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_6119_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V3: product_prod_nat_nat] : Xa = some_P7363390416028606310at_nat(V3)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xa = some_P7363390416028606310at_nat(A4) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb = some_P7363390416028606310at_nat(B3) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_6120_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_num_fun_num_num,Xa: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa,Xb) = Y )
     => ( ( ( Xa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V3: num] : Xa = some_num(V3)
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A4: num] :
                ( ( Xa = some_num(A4) )
               => ! [B3: num] :
                    ( ( Xb = some_num(B3) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_6121_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa),Xb) = Y )
     => ( ( ( Xa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V3: nat] : Xa = some_nat(V3)
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A4: nat] :
                ( ( Xa = some_nat(A4) )
               => ! [B3: nat] :
                    ( ( Xb = some_nat(B3) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_6122_option_Osize_I3_J,axiom,
    aa_option_nat_nat(size_size_option_nat,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6123_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6124_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6125_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D3: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D3))
    <=> ( ~ 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),D3)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_6126_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D3: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D3)))
    <=> ( ~ 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),D3)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_6127_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D3: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D3)))
    <=> ( ~ 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),D3)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_6128_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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_or1269000886237332187st_nat(C),D3)))
    <=> ( ~ 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),D3)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_6129_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D3)))
    <=> ( ~ 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),D3)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_6130_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D3)))
    <=> ( ~ 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),D3)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_6131_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
     => ( ( Mi != Ma )
       => ( aa_option_nat_nat(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,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_6132_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_6133_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_6134__C5_Oprems_C,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),x))) ).

% "5.prems"
tff(fact_6135_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,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_6136_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_delete(T,X)),N)) ) ).

% delete_pres_valid
tff(fact_6137_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_6138_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_6139_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_6140_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_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_6141_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_6142_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_6143_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(vEBT_vebt_member(T),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_6144_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% maxt_member
tff(fact_6145_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( pp(aa_nat_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_6146_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6147_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6148_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,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_6149_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
          & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_6150_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
          & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_6151_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
          & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_6152_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
          & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_6153_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
          & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_6154_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_6155_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(L),I))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_6156_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or633870826150836451st_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_eq_rat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_6157_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or7049704709247886629st_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_eq_num,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_6158_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_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_eq_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_6159_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_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_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_6160_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or1222579329274155063t_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_eq_real,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_6161_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_6162_aset_I2_J,axiom,
    ! [D5: int,A3: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A3))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A3))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X5))
                | pp(aa_int_bool(Q,X5)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5))) ) ) ) ) ) ).

% aset(2)
tff(fact_6163_aset_I1_J,axiom,
    ! [D5: int,A3: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A3))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A3))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X5))
                & pp(aa_int_bool(Q,X5)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5))) ) ) ) ) ) ).

% aset(1)
tff(fact_6164_bset_I2_J,axiom,
    ! [D5: int,B4: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B4))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B4))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B4))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X5))
                | pp(aa_int_bool(Q,X5)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5))) ) ) ) ) ) ).

% bset(2)
tff(fact_6165_bset_I1_J,axiom,
    ! [D5: int,B4: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B4))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B4))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B4))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X5))
                & pp(aa_int_bool(Q,X5)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5))) ) ) ) ) ) ).

% bset(1)
tff(fact_6166_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M7: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M7)) )
       => ~ ! [M4: nat] :
              ( pp(aa_nat_bool(P,M4))
             => ~ ! [X5: nat] :
                    ( pp(aa_nat_bool(P,X5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),M4)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_6167_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_6168_bset_I9_J,axiom,
    ! [D3: int,D5: int,B4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B4))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5)),T))) ) ) ) ).

% bset(9)
tff(fact_6169_bset_I10_J,axiom,
    ! [D3: int,D5: int,B4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B4))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5)),T))) ) ) ) ).

% bset(10)
tff(fact_6170_aset_I9_J,axiom,
    ! [D3: int,D5: int,A3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A3))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5)),T))) ) ) ) ).

% aset(9)
tff(fact_6171_aset_I10_J,axiom,
    ! [D3: int,D5: int,A3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A3))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5)),T))) ) ) ) ).

% aset(10)
tff(fact_6172_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_6173_periodic__finite__ex,axiom,
    ! [D3: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3)))) )
       => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D3)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_6174_aset_I7_J,axiom,
    ! [D5: int,A3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A3))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5))) ) ) ) ).

% aset(7)
tff(fact_6175_aset_I5_J,axiom,
    ! [D5: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),A3))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5)),T)) ) ) ) ) ).

% aset(5)
tff(fact_6176_aset_I4_J,axiom,
    ! [D5: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),A3))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X5 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5) != T ) ) ) ) ) ).

% aset(4)
tff(fact_6177_aset_I3_J,axiom,
    ! [D5: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A3))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X5 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5) = T ) ) ) ) ) ).

% aset(3)
tff(fact_6178_bset_I7_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),B4))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B4))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5))) ) ) ) ) ).

% bset(7)
tff(fact_6179_bset_I5_J,axiom,
    ! [D5: int,B4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B4))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5)),T)) ) ) ) ).

% bset(5)
tff(fact_6180_bset_I4_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(T),B4))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B4))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X5 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5) != T ) ) ) ) ) ).

% bset(4)
tff(fact_6181_bset_I3_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B4))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B4))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X5 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5) = T ) ) ) ) ) ).

% bset(3)
tff(fact_6182_bset_I6_J,axiom,
    ! [D5: int,B4: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),B4))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5)),T)) ) ) ) ).

% bset(6)
tff(fact_6183_bset_I8_J,axiom,
    ! [D5: int,T: int,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B4))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),B4))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D5))) ) ) ) ) ).

% bset(8)
tff(fact_6184_aset_I6_J,axiom,
    ! [D5: int,T: int,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A3))
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
               => ! [Xb3: int] :
                    ( pp(aa_set_int_bool(member_int(Xb3),A3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5)),T)) ) ) ) ) ).

% aset(6)
tff(fact_6185_aset_I8_J,axiom,
    ! [D5: int,A3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D5)))
             => ! [Xb3: int] :
                  ( pp(aa_set_int_bool(member_int(Xb3),A3))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D5))) ) ) ) ).

% aset(8)
tff(fact_6186_cppi,axiom,
    ! [D5: int,P: fun_int_bool,P2: fun_int_bool,A3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(P2,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
                 => ! [Xb2: int] :
                      ( pp(aa_set_int_bool(member_int(Xb2),A3))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D5))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa_int_bool(P2,X4))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),A3))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_6187_cpmi,axiom,
    ! [D5: int,P: fun_int_bool,P2: fun_int_bool,B4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D5))
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(P2,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D5)))
                 => ! [Xb2: int] :
                      ( pp(aa_set_int_bool(member_int(Xb2),B4))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D5))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa_int_bool(P2,X4))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D5)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D5)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),B4))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_6188_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D3: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D3))
    <=> ( ( ~ 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),D3))
            & ( pp(aa_set_nat_bool(ord_less_set_nat(C),A))
              | pp(aa_set_nat_bool(ord_less_set_nat(B),D3)) ) ) )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D3)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_6189_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D3: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D3)))
    <=> ( ( ~ 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),D3))
            & ( 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),D3)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D3)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_6190_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D3: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D3)))
    <=> ( ( ~ 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),D3))
            & ( 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),D3)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D3)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_6191_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D3: 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),D3)))
    <=> ( ( ~ 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),D3))
            & ( 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),D3)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D3)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_6192_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D3: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D3)))
    <=> ( ( ~ 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),D3))
            & ( 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),D3)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D3)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_6193_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D3: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D3)))
    <=> ( ( ~ 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),D3))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D3)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D3)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_6194_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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,aa_num_num(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))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_6195_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
     => ( pp(aa_nat_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,aa_num_num(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))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_6196_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))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_6197_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))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_6198_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_6199_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_6200_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,aa_num_num(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_num_num(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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_6201_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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_6202__C5_OIH_C_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X5),na))
        & ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(X5,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(X5),x)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ) ) ).

% "5.IH"(1)
tff(fact_6203_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),X) = 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_6204__C5_OIH_C_I2_J,axiom,
    ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(summary,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(summary),x)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% "5.IH"(2)
tff(fact_6205_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_6206_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A5: bool,B5: bool] : T = vEBT_Leaf(A5,B5) ) ).

% deg1Leaf
tff(fact_6207_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A4: bool,B3: bool] : T = vEBT_Leaf(A4,B3) ) ).

% deg_1_Leaf
tff(fact_6208_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A4: bool,B3: bool] : T = vEBT_Leaf(A4,B3) ) ) ).

% deg_1_Leafy
tff(fact_6209_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_6210_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_6211_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_6212_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).

% semiring_norm(89)
tff(fact_6213_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_6214_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_6215_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,aa_num_num(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(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,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% tdeletemimi
tff(fact_6216_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_6217_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_6218_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)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(9)
tff(fact_6219_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_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_6220_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_6221_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_6222_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_6223_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_6224_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)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_6225_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_6226_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)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(81)
tff(fact_6227_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_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_6228_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_6229_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_6230_or__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X)) ).

% or_numerals(8)
tff(fact_6231_or__numerals_I8_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))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_numerals(8)
tff(fact_6232_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% or_numerals(2)
tff(fact_6233_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_numerals(2)
tff(fact_6234_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_6235_semiring__norm_I3_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_6236_semiring__norm_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% semiring_norm(4)
tff(fact_6237_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_6238_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_6239_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)) = aa_num_num(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_6240_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_6241_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_6242_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_6243_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_6244_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)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)))) ).

% semiring_norm(16)
tff(fact_6245_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_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_6246_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)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(74)
tff(fact_6247_or__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X)) ).

% or_numerals(5)
tff(fact_6248_or__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_numerals(5)
tff(fact_6249_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% or_numerals(1)
tff(fact_6250_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_numerals(1)
tff(fact_6251_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% xor_numerals(1)
tff(fact_6252_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_numerals(1)
tff(fact_6253_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y)) ).

% xor_numerals(2)
tff(fact_6254_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_numerals(2)
tff(fact_6255_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X)) ).

% xor_numerals(5)
tff(fact_6256_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_numerals(5)
tff(fact_6257_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,X)) ).

% xor_numerals(8)
tff(fact_6258_xor__numerals_I8_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))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_numerals(8)
tff(fact_6259_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: 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,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_6260_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J: 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,J),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_6261_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: 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,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_6262_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: 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,J),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_6263_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_6264_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_6265_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_6266_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_6267_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_6268_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_6269_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% xor_numerals(7)
tff(fact_6270_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_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(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% xor_numerals(7)
tff(fact_6271_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_6272_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_6273_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,aa_num_num(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_6274_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,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_6275_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,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_6276_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,aa_num_num(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_6277_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,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_6278_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,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_6279_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(4)
tff(fact_6280_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(4)
tff(fact_6281_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(6)
tff(fact_6282_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(6)
tff(fact_6283_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se1409905431419307370or_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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(7)
tff(fact_6284_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(7)
tff(fact_6285_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(4)
tff(fact_6286_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_6287_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(6)
tff(fact_6288_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_6289_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,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_6290_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_6291_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6292_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_6293_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_6294_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_6295_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_6296_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_6297_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_6298_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_6299_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N2))
       => ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M4: num] : X != product_Pair_num_num(aa_num_num(bit0,M4),one)
           => ( ! [M4: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N2))
             => ( ! [M4: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N2))
               => ( ! [M4: num] : X != product_Pair_num_num(aa_num_num(bit1,M4),one)
                 => ( ! [M4: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N2))
                   => ~ ! [M4: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_6300_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] : Y != aa_num_num(bit0,X22)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_6301_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_6302_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_6303_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),X4)
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S3),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_6304_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_6305_numeral__Bit1,axiom,
    ! [N: num] : aa_num_real(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,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_Bit1
tff(fact_6306_numeral__Bit1,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_Bit1
tff(fact_6307_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_6308_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_6309_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_6310_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,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_6311_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != 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,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6312_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6313_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6314_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: 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,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6315_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: 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,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6316_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6317_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: 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,aa_num_num(bit0,Q3))) = 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,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6318_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: 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,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6319_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6320_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_6321_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_6322_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_6323_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6324_power__minus__Bit1,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(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_6325_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_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Uu2) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_6326_set__update__memI,axiom,
    ! [N: nat,Xs: list_option_nat,X: option_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6086282163384603972on_nat(Xs)))
     => member_option_nat(X,set_option_nat2(list_u3411377215356412978on_nat(Xs,N,X))) ) ).

% set_update_memI
tff(fact_6327_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_6328_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(X,set_set_nat2(list_update_set_nat(Xs,N,X))) ) ).

% set_update_memI
tff(fact_6329_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)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_6330_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_6331_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_6332_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_6333_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_6334_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X: bool,J: 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)),J))
      <=> ( ( ( I = J )
           => pp(X) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_6335_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_6336_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_6337_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_6338_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_6339_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_6340_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_6341_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_6342_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_6343_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,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit1_div_2
tff(fact_6344_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit1_div_2
tff(fact_6345_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_6346_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_6347_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_6348_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: 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,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6349_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6350_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6351_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_6352_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_6353_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_6354_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_6355_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(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_6356_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_6357_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_6358_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_6359_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,B3: bool,Va: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb)
           => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(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),X4) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_6360_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,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,B3),zero_zero_nat)
     => ( ! [Uv: bool,Uw: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N2))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(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),X4) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_6361_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,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),X4)
         => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(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),X4) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_6362_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,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),X4)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),X4)
       => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),X4)
         => ( ! [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(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),X4) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_6363_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_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),zero_zero_nat)
     => ( ! [A4: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: bool,B3: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),aa_nat_nat(suc,aa_nat_nat(suc,N2)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2),Uu)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary2),X4)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary2),X4)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(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),X4) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.cases
tff(fact_6364_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,Uv: bool,Uw: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Uw)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb),X4)
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),X4)
           => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_6365_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: 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,aa_num_num(bit0,Q3))) = 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,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6366_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6367_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6368_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: 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,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6369_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6370_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6371_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_6372_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_6373_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_6374_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_6375_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_6376_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_6377_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,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_6378_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_6379_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(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_6380_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_6381_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_6382_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( pp(A4)
                 => ( Y = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A4)
                 => ( ( pp(B3)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_6383_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( pp(B3)
                 => ( Y = some_nat(one_one_nat) ) )
                & ( ~ pp(B3)
                 => ( ( pp(A4)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A4)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_6384_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(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_6385_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,aa_num_num(bit0,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_6386_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry),X) = 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_6387_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A4: bool,B3: bool] : A1 = vEBT_Leaf(A4,B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                     => ( ( M4 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                         => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                       => ( ( M4 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                           => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_13))
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                                   => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                         => ( ( M4 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                             => ( ! [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,aa_num_num(bit0,one))),M4)))
                                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_12))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I2)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                                       => ~ ? [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,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,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [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,aa_num_num(bit0,one))),M4)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I2 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X5: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N2) = I2 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X5))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X5),N2)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M4))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                           => ( ( M4 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M4) )
                               => ( ! [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,aa_num_num(bit0,one))),M4)))
                                     => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_12))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I2)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                                         => ~ ? [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,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,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [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,aa_num_num(bit0,one))),M4)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I2 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X5: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N2) = I2 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X5))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_6388_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
    <=> ( ( ? [A5: bool,B5: bool] : A1 = vEBT_Leaf(A5,B5)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N3))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N3)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N3))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3) )
            & ! [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,aa_num_num(bit0,one))),N3)))
               => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_12))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [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,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,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [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,aa_num_num(bit0,one))),N3)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N3) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N3)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N3)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N3)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) )
            & ! [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,aa_num_num(bit0,one))),aa_nat_nat(suc,N3))))
               => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_12))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [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,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,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [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,aa_num_num(bit0,one))),aa_nat_nat(suc,N3))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N3) = I4 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N3))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I4 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_6389_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,aa_num_num(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,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),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,aa_num_num(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,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_6390_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,aa_num_num(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,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),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,aa_num_num(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,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_6391_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% member_bound_size_univ
tff(fact_6392_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_6393_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% succ_bound_size_univ
tff(fact_6394_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% pred_bound_size_univ
tff(fact_6395_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_6396_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_6397_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_6398_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_6399_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_6400_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_6401_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_6402_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(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)))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_6403_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(1)
tff(fact_6404_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(1)
tff(fact_6405_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(1)
tff(fact_6406_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_6407_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(aa_int_fun_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(aa_int_fun_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_6408_max__0__1_I3_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_6409_max__0__1_I3_J,axiom,
    ! [X: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(3)
tff(fact_6410_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(3)
tff(fact_6411_max__0__1_I3_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(3)
tff(fact_6412_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_6413_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_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_6414_max__0__1_I4_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_6415_max__0__1_I4_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(X) ).

% max_0_1(4)
tff(fact_6416_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),zero_zero_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(4)
tff(fact_6417_max__0__1_I4_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(4)
tff(fact_6418_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_6419_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_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_6420_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_6421_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_6422_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_6423_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_6424_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_6425_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_6426_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_6427_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_6428_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_6429_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_6430_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_6431_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_6432_max__0__1_I5_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_6433_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_Code_integer(one_one_Code_integer,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(5)
tff(fact_6434_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(5)
tff(fact_6435_max__0__1_I5_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(5)
tff(fact_6436_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_6437_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_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_6438_max__0__1_I6_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_6439_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),one_one_Code_integer) = numera6620942414471956472nteger(X) ).

% max_0_1(6)
tff(fact_6440_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),one_one_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(6)
tff(fact_6441_max__0__1_I6_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(6)
tff(fact_6442_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_6443_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_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_6444_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(2)
tff(fact_6445_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_6446_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(2)
tff(fact_6447_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(aa_int_fun_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(aa_int_fun_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_6448_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,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(3)
tff(fact_6449_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_6450_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(3)
tff(fact_6451_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(aa_int_fun_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(aa_int_fun_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_6452_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,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(4)
tff(fact_6453_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(4)
tff(fact_6454_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(4)
tff(fact_6455_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(aa_int_fun_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(aa_int_fun_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_6456_max__def,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ) ).

% max_def
tff(fact_6457_max__def,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => ( ord_max_Code_integer(A,B) = B ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => ( ord_max_Code_integer(A,B) = A ) ) ) ).

% max_def
tff(fact_6458_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ord_max_set_nat(A,B) = B ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ord_max_set_nat(A,B) = A ) ) ) ).

% max_def
tff(fact_6459_max__def,axiom,
    ! [A: rat,B: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = B ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ) ).

% max_def
tff(fact_6460_max__def,axiom,
    ! [A: num,B: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ) ).

% max_def
tff(fact_6461_max__def,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(ord_max_nat,A),B) = B ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ) ).

% max_def
tff(fact_6462_max__def,axiom,
    ! [A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ) ).

% max_def
tff(fact_6463_max__absorb1,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Y),X))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6464_max__absorb1,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X))
     => ( ord_max_Code_integer(X,Y) = X ) ) ).

% max_absorb1
tff(fact_6465_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( ord_max_set_nat(X,Y) = X ) ) ).

% max_absorb1
tff(fact_6466_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(ord_max_rat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_6467_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6468_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6469_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_6470_max__absorb2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X),Y))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6471_max__absorb2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
     => ( ord_max_Code_integer(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_6472_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( ord_max_set_nat(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_6473_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(ord_max_rat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6474_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6475_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6476_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_6477_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_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_6478_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_rat_rat(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_6479_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_int_int(aa_int_fun_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_6480_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = ord_max_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X),aa_int_Code_integer(ring_18347121197199848620nteger,Y)) ).

% of_int_max
tff(fact_6481_of__nat__max,axiom,
    ! [X: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,semiri4216267220026989637d_enat(X)),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_6482_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,X),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_6483_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_6484_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_rat_rat(ord_max_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(semiri681578069525770553at_rat,Y)) ).

% of_nat_max
tff(fact_6485_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_6486_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(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_6487_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_6488_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_6489_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_6490_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_6491_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(ord_max_Code_integer(X,Y),Z) = ord_max_Code_integer(plus_p5714425477246183910nteger(X,Z),plus_p5714425477246183910nteger(Y,Z)) ).

% max_add_distrib_left
tff(fact_6492_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_6493_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(X),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_rat_rat(plus_plus_rat(X),Z)) ).

% max_add_distrib_right
tff(fact_6494_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_6495_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% max_add_distrib_right
tff(fact_6496_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(X,ord_max_Code_integer(Y,Z)) = ord_max_Code_integer(plus_p5714425477246183910nteger(X,Y),plus_p5714425477246183910nteger(X,Z)) ).

% max_add_distrib_right
tff(fact_6497_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(X,Y),Z) = ord_max_Code_integer(minus_8373710615458151222nteger(X,Z),minus_8373710615458151222nteger(Y,Z)) ).

% max_diff_distrib_left
tff(fact_6498_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_6499_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_6500_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_6501_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: 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),Q3)) = 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),Q3)) ).

% nat_add_max_right
tff(fact_6502_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: 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)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q3)) ).

% nat_add_max_left
tff(fact_6503_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: 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)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_max_left
tff(fact_6504_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: 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),Q3)) = 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),Q3)) ).

% nat_mult_max_right
tff(fact_6505_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_6506_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_6507_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(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_6508_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_6509_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(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_6510_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_6511_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(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_6512_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(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_6513_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(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_6514_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_6515_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,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_6516_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(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_6517_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,aa_num_num(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_6518_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_6519_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_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_6520_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))) ) ).

% pred_bound_height
tff(fact_6521_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,aa_num_num(bit1,one)))))))) ) ).

% succ_bound_height
tff(fact_6522_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6523_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb3
tff(fact_6524_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_6525_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_6526_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_6527_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_6528_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_6529_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_6530_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A),B))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb4
tff(fact_6531_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_6532_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_6533_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_6534_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_6535_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_6536_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_6537_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_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_6538_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_6539_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_6540_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_6541_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_6542_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_6543_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(aa_int_fun_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_6544_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb1
tff(fact_6545_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb1
tff(fact_6546_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_6547_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_6548_max_Oabsorb1,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(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_6549_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_6550_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb2
tff(fact_6551_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb2
tff(fact_6552_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_6553_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_6554_max_Oabsorb2,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(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_6555_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_6556_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
        & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).

% max.bounded_iff
tff(fact_6557_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A)) ) ) ).

% max.bounded_iff
tff(fact_6558_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),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_eq_rat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_6559_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_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_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.bounded_iff
tff(fact_6560_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_6561_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
    <=> ( 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)) ) ) ).

% max.bounded_iff
tff(fact_6562_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_6563_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q3),zero_z5237406670263579293d_enat) = Q3 ).

% max_enat_simps(2)
tff(fact_6564_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),B))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.coboundedI2
tff(fact_6565_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.coboundedI2
tff(fact_6566_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: 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,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI2
tff(fact_6567_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI2
tff(fact_6568_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: 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,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_6569_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: 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,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI2
tff(fact_6570_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.coboundedI1
tff(fact_6571_max_OcoboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.coboundedI1
tff(fact_6572_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( 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,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI1
tff(fact_6573_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( 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,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI1
tff(fact_6574_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( 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,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_6575_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( 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,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI1
tff(fact_6576_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_6577_max_Oabsorb__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
    <=> ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_6578_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_6579_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_6580_max_Oabsorb__iff2,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(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_6581_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_6582_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_6583_max_Oabsorb__iff1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
    <=> ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_6584_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_6585_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_6586_max_Oabsorb__iff1,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(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_6587_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_6588_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),X))
        | pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6589_le__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Z),ord_max_Code_integer(X,Y)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Z),X))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6590_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6591_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_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_eq_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6592_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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_eq_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6593_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_6594_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ).

% max.cobounded2
tff(fact_6595_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),ord_max_Code_integer(A,B))) ).

% max.cobounded2
tff(fact_6596_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded2
tff(fact_6597_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded2
tff(fact_6598_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded2
tff(fact_6599_max_Ocobounded2,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded2
tff(fact_6600_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ).

% max.cobounded1
tff(fact_6601_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),ord_max_Code_integer(A,B))) ).

% max.cobounded1
tff(fact_6602_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded1
tff(fact_6603_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded1
tff(fact_6604_max_Ocobounded1,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(ord_max_nat,A),B))) ).

% max.cobounded1
tff(fact_6605_max_Ocobounded1,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded1
tff(fact_6606_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
    <=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.order_iff
tff(fact_6607_max_Oorder__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
    <=> ( A = ord_max_Code_integer(A,B) ) ) ).

% max.order_iff
tff(fact_6608_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_6609_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_6610_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_6611_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_6612_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_6613_max_OboundedI,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),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A)) ) ) ).

% max.boundedI
tff(fact_6614_max_OboundedI,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),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_6615_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( 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,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_6616_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_6617_max_OboundedI,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),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_6618_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).

% max.boundedE
tff(fact_6619_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A))
     => ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
         => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A)) ) ) ).

% max.boundedE
tff(fact_6620_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),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_eq_rat,C),A)) ) ) ).

% max.boundedE
tff(fact_6621_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_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_eq_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.boundedE
tff(fact_6622_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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_eq_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.boundedE
tff(fact_6623_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
     => ~ ( 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)) ) ) ).

% max.boundedE
tff(fact_6624_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A)) ) ).

% max.orderI
tff(fact_6625_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = ord_max_Code_integer(A,B) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).

% max.orderI
tff(fact_6626_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% max.orderI
tff(fact_6627_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% max.orderI
tff(fact_6628_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% max.orderI
tff(fact_6629_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% max.orderI
tff(fact_6630_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.orderE
tff(fact_6631_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( A = ord_max_Code_integer(A,B) ) ) ).

% max.orderE
tff(fact_6632_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_6633_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_6634_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_6635_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_6636_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D3: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(D3),B))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,C),D3)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ) ).

% max.mono
tff(fact_6637_max_Omono,axiom,
    ! [C: code_integer,A: code_integer,D3: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,D3),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(C,D3)),ord_max_Code_integer(A,B))) ) ) ).

% max.mono
tff(fact_6638_max_Omono,axiom,
    ! [C: rat,A: rat,D3: rat,B: rat] :
      ( 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,D3),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D3)),aa_rat_rat(ord_max_rat(A),B))) ) ) ).

% max.mono
tff(fact_6639_max_Omono,axiom,
    ! [C: num,A: num,D3: num,B: num] :
      ( 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,D3),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D3)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ) ).

% max.mono
tff(fact_6640_max_Omono,axiom,
    ! [C: nat,A: nat,D3: nat,B: nat] :
      ( 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,D3),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D3)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_6641_max_Omono,axiom,
    ! [C: int,A: int,D3: int,B: int] :
      ( 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,D3),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D3)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).

% max.mono
tff(fact_6642_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),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_6643_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_6644_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_6645_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),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_6646_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_6647_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_6648_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(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_6649_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),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_6650_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_6651_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_6652_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),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_6653_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_6654_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_6655_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(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_6656_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_6657_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_6658_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_6659_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_6660_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_6661_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_6662_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(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_6663_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_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_6664_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_6665_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_6666_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_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_6667_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_6668_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_6669_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(aa_int_fun_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_6670_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(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_6671_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_6672_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_6673_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(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_6674_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_6675_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_6676_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(aa_int_fun_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_6677_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_6678_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))))) ) ) ).

% insert_bound_size_univ
tff(fact_6679_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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => ( vEBT_vebt_pred(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_6680_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,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( vEBT_vebt_pred(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% pred_lesseq_max
tff(fact_6681_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,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( vEBT_vebt_succ(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% succ_greatereq_min
tff(fact_6682_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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => ( vEBT_vebt_succ(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_6683_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_6684_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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),aa_option_nat_nat(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)),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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(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_6685_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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_6686_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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_6687_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,aa_num_num(bit0,one))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_6688_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,aa_num_num(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_num_num(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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(the_nat,vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_6689_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,aa_num_num(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_num_num(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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,aa_option_nat_nat(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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_6690_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,aa_num_num(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa_option_nat_nat(the_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),aa_option_nat_nat(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)),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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(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_6691_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,aa_num_num(bit0,one))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = 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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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(aa_boo8970175072276269406l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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(aa_boo8970175072276269406l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_in_range
tff(fact_6692_max__def__raw,axiom,
    ! [X5: extended_enat,Xa2: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X5),Xa2))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X5),Xa2))
       => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6693_max__def__raw,axiom,
    ! [X5: code_integer,Xa2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X5),Xa2))
       => ( ord_max_Code_integer(X5,Xa2) = Xa2 ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X5),Xa2))
       => ( ord_max_Code_integer(X5,Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6694_max__def__raw,axiom,
    ! [X5: set_nat,Xa2: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X5),Xa2))
       => ( ord_max_set_nat(X5,Xa2) = Xa2 ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(X5),Xa2))
       => ( ord_max_set_nat(X5,Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6695_max__def__raw,axiom,
    ! [X5: rat,Xa2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa2))
       => ( aa_rat_rat(ord_max_rat(X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa2))
       => ( aa_rat_rat(ord_max_rat(X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6696_max__def__raw,axiom,
    ! [X5: num,Xa2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa2))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa2))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6697_max__def__raw,axiom,
    ! [X5: nat,Xa2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6698_max__def__raw,axiom,
    ! [X5: int,Xa2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_6699_lambda__zero,axiom,
    aTP_Lamm_af = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_6700_lambda__zero,axiom,
    aTP_Lamm_ag = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_6701_lambda__zero,axiom,
    aTP_Lamm_ah = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_6702_lambda__zero,axiom,
    aTP_Lamm_ai = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_6703_numeral__code_I2_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_code(2)
tff(fact_6704_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_code(2)
tff(fact_6705_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_code(2)
tff(fact_6706_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(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_6707_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_6708_nat__less__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ) ).

% nat_less_as_int
tff(fact_6709_nat__leq__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ) ).

% nat_leq_as_int
tff(fact_6710_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_6711_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_real(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,aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_code(3)
tff(fact_6712_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_code(3)
tff(fact_6713_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_6714_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_6715_power__numeral__even,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(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_6716_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,aa_num_num(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_6717_power__numeral__even,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(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_6718_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,aa_num_num(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_6719_power__numeral__even,axiom,
    ! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_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_6720_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_6721_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_6722_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_6723_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_6724_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(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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_6725_nat__plus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_6726_nat__times__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_times_as_int
tff(fact_6727_nat__minus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_minus_as_int
tff(fact_6728_nat__div__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_6729_nat__mod__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_mod_as_int
tff(fact_6730_diff__nat__eq__if,axiom,
    ! [Z4: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4))) ) ) ) ).

% diff_nat_eq_if
tff(fact_6731_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(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_6732_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(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_6733_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_6734_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(vEBT_vebt_member(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_6735_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa))
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => pp(A4) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => pp(B3) )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = 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)
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_6736_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(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(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,aa_num_num(bit0,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))),one_one_nat)))))))))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_6737_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa))
      <=> pp(Y) )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => pp(Y) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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)
                    <=> ~ ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_6738_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa))
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( ( Xa = zero_zero_nat )
               => pp(A4) )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => pp(B3) )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
         => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
           => ( ! [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = 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)
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_6739_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(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_6740_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( pp(aa_nat_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_6741_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_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_6742_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(X),Xa) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = 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(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),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,Xa),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),Xa)),one_one_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(bit0,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(bit0,one))))))),one_one_nat)))))))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_6743_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(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(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_6744_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(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(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_6745_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(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) = 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(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) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(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_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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(aa_boo8970175072276269406l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),X)),aa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a_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))),Summary)),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_6746_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( Xa = zero_zero_nat )
             => ( Y != vEBT_Leaf(fFalse,B3) ) ) )
       => ( ! [A4: bool] :
              ( ? [B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf(A4,fFalse) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ? [N2: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
                 => ( Y != vEBT_Leaf(A4,B3) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                   => ( Y != 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 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                     => ( Y != 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 = 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,Xa),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( Y = 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,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y = 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t_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(bit0,one)))))),Summary2)),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_6747_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa) = Y )
     => ( ! [Uu: bool,B3: bool] :
            ( ( X = vEBT_Leaf(Uu,B3) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( pp(B3)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: bool,Uw: bool] : X = vEBT_Leaf(Uv,Uw)
           => ( ? [N2: nat] : Xa = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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,Xa),Mi2))
                           => ( Y = some_nat(Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),some_nat(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,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))),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,aa_num_num(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,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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,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,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary2,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,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_6748_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa) = Y )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( ( Xa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: bool] :
              ( ? [Uw: bool] : X = vEBT_Leaf(A4,Uw)
             => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A4)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A4)
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ? [Va: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ~ ( ( pp(B3)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( ( pp(A4)
                           => ( Y = some_nat(zero_zero_nat) ) )
                          & ( ~ pp(A4)
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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),Xa))
                             => ( Y = some_nat(Ma2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( 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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),some_nat(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,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))),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,aa_num_num(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,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,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa),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,aa_num_num(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,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,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,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary2,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,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_6749_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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,aa_num_num(bit0,one)))))))) ) ).

% insert_bound_height
tff(fact_6750_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B3) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y = vEBT_Leaf(A4,fTrue) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y = vEBT_Leaf(A4,B3) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts,S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) ) )
           => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = 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 != 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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),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,Xa),Mi2)),Xa),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,Xa),Mi2)),Mi2),Xa)),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),Summary2)),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_6751_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(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_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,aa_num_num(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)))),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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),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,aa_num_num(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,aa_num_num(bit0,one))))),Summary)),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_6752_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,aa_num_num(bit0,one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_6753_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,aa_num_num(bit0,one)))),zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_6754_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,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_6755_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) = aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_6756_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) = aa_rat_rat(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) = aa_rat_rat(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,aa_num_num(bit0,one)))),zero_zero_int),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_6757_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(X),Xa) = Y )
     => ( ( ? [Uu: bool,B3: bool] : X = vEBT_Leaf(Uu,B3)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: bool,Uw: bool] : X = vEBT_Leaf(Uv,Uw)
           => ( ? [N2: nat] : Xa = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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,Xa),Mi2))
                           => ( Y = one_one_nat ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_6758_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(X),Xa) = Y )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,Uw: bool] : X = vEBT_Leaf(A4,Uw)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ? [Va: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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),Xa))
                             => ( Y = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( 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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_6759_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_6760_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_6761_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_6762_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(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_6763_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(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_6764_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(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_6765_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(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_6766_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6767_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6768_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts2,S),X) = vEBT_Node(Info,zero_zero_nat,Ts2,S) ).

% vebt_insert.simps(2)
tff(fact_6769_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ'
tff(fact_6770_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ'
tff(fact_6771_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ).

% vebt_insert.simps(3)
tff(fact_6772_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_6773_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(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(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_6774_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(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(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_6775_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(X),Xa) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( 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,Xa),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 = vEBT_Node(Info2,zero_zero_nat,Ts,S3)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = 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,aa_num_num(bit0,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_6776_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(X),Xa) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != one_one_nat ) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),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),Xa)),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),Xa)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_6777_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(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,aa_num_num(bit0,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),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,aa_num_num(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,aa_num_num(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,aa_num_num(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_6778_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(X),Xa) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero_nat,Ts,S3)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = 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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_6779_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(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,aa_num_num(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,aa_num_num(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,aa_num_num(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_6780_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,vEBT_T_m_i_n_N_u_l_l(T)),one_one_nat)) ).

% minNull_bound
tff(fact_6781_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(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_6782_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(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_6783_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( pp(aa_nat_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_t2(T),Y)),one_one_nat)) ) ) ).

% insersimp'
tff(fact_6784_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_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_t2(T),L)),one_one_nat)) ) ) ).

% insertsimp'
tff(fact_6785_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,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_6786_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6787_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(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_6788_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6789_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(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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_6790_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(X),Xa) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ? [N2: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = 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 = 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 = 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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_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(bit0,one)))))))),one_one_nat)))))),one_one_nat)))))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_6791_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa))
      <=> pp(Y) )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb)
               => ( pp(Y)
                <=> ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_6792_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa))
     => ( ! [Uu: bool,Uv: bool] : X != vEBT_Leaf(Uu,Uv)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb)
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2)
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
             => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_6793_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa))
      <=> pp(Y) )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_6794_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa))
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => pp(A4) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => pp(B3) )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_6795_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_6796_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_6797_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(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_6798_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_6799_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_6800_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_6801_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_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_6802_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(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_6803_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,vEBT_T_m_a_x_t(T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).

% maxt_bound
tff(fact_6804_mint__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,vEBT_T_m_i_n_t(T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).

% mint_bound
tff(fact_6805_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] : 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_6806_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(X) = Y )
     => ( ! [A4: bool] :
            ( ? [B3: bool] : X = vEBT_Leaf(A4,B3)
           => ( 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 = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_6807_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_6808_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_6809_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(vEBT_VEBT_membermima(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
           => pp(aa_nat_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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_6810_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(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,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))),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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,aa_num_num(bit0,one)))),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_6811_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(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,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(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,aa_num_num(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),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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_6812_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(X),Xa) = Y )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,Uw: bool] : X = vEBT_Leaf(A4,Uw)
           => ( ( Xa = 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,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ? [Va: nat] : Xa = 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(B3),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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),Xa)),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(bit0,aa_num_num(bit1,aa_num_num(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,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,aa_num_num(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,aa_num_num(bit0,one))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(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,aa_num_num(bit0,aa_num_num(bit0,one)))),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary2,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,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_6813_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(X),Xa) = Y )
     => ( ( ? [Uu: bool,B3: bool] : X = vEBT_Leaf(Uu,B3)
         => ( ( Xa = 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] : Xa = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V3),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 = 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,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(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,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,aa_num_num(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),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary2,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,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_6814_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb)
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2)
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
         => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_6815_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(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,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,X),Mi)),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,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,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma))),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,aa_num_num(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,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary)))))),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,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_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_d_e_l_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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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),vEBT_T_m_i_n_N_u_l_l(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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_plu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ion_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_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,aa_num_num(bit0,aa_num_num(bit1,one)))),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_option_nat_nat(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,aa_num_num(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_6816_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa))
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( ( Xa = zero_zero_nat )
               => pp(A4) )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => pp(B3) )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_6817_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [N2: nat] :
                      ( ( Xa = 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,B3),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2,Summary2),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = 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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2,Summary2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = 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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_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(bit0,one)))))))),one_one_nat)))))),one_one_nat)))))) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_6818_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu,B3) )
             => ( ( Xa = 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,B3),zero_zero_nat)) ) ) )
         => ( ! [Uv: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv,Uw) )
               => ! [N2: nat] :
                    ( ( Xa = 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))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),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(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = 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,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(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,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,aa_num_num(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),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary2,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,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_6819_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( Xa = 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) )
               => ( ( Xa = 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,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [Va: nat] :
                      ( ( Xa = 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(B3),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,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2),Xa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),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(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = 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),Xa)),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(bit0,aa_num_num(bit1,aa_num_num(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,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,aa_num_num(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,aa_num_num(bit0,one))),vEBT_T_m_i_n_t(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(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,aa_num_num(bit0,aa_num_num(bit0,one)))),vEBT_T_m_a_x_t(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary2,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,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_6820_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aj(T),X)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_6821_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ak(T),X)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_6822_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_6823_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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_6824_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_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_6825_max__bot,axiom,
    ! [X: set_Extended_enat] : ord_ma4205026669011143323d_enat(bot_bo7653980558646680370d_enat,X) = X ).

% max_bot
tff(fact_6826_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_6827_max__bot,axiom,
    ! [X: set_nat] : ord_max_set_nat(bot_bot_set_nat,X) = X ).

% max_bot
tff(fact_6828_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_6829_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_6830_max__bot,axiom,
    ! [X: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,bot_bo4199563552545308370d_enat),X) = X ).

% max_bot
tff(fact_6831_max__bot2,axiom,
    ! [X: set_Extended_enat] : ord_ma4205026669011143323d_enat(X,bot_bo7653980558646680370d_enat) = X ).

% max_bot2
tff(fact_6832_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_6833_max__bot2,axiom,
    ! [X: set_nat] : ord_max_set_nat(X,bot_bot_set_nat) = X ).

% max_bot2
tff(fact_6834_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_6835_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_6836_max__bot2,axiom,
    ! [X: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),bot_bo4199563552545308370d_enat) = X ).

% max_bot2
tff(fact_6837_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B) = bot_bo7653980558646680370d_enat )
    <=> ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6838_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6839_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6840_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6841_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6842_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6843_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_6844_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = set_or5403411693681687835d_enat(A,B) )
    <=> ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6845_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6846_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6847_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6848_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6849_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6850_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_6851_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
     => ( set_or5403411693681687835d_enat(A,B) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_6852_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_6853_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_6854_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_6855_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_6856_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_6857_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat),A)) ).

% bot.extremum
tff(fact_6858_bot_Oextremum,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A)) ).

% bot.extremum
tff(fact_6859_bot_Oextremum,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% bot.extremum
tff(fact_6860_bot_Oextremum,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(bot_bot_set_nat),A)) ).

% bot.extremum
tff(fact_6861_bot_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A)) ).

% bot.extremum
tff(fact_6862_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat))
    <=> ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
tff(fact_6863_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_6864_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_6865_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_6866_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_6867_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat))
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
tff(fact_6868_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_6869_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_6870_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),bot_bot_set_nat))
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_6871_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_6872_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(bot_bo7653980558646680370d_enat),A)) ) ).

% bot.not_eq_extremum
tff(fact_6873_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A)) ) ).

% bot.not_eq_extremum
tff(fact_6874_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_set_nat(bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_6875_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A)) ) ).

% bot.not_eq_extremum
tff(fact_6876_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_6877_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] : ~ pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A),bot_bo7653980558646680370d_enat)) ).

% bot.extremum_strict
tff(fact_6878_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_6879_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_6880_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_6881_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_6882_diff__shunt__var,axiom,
    ! [X: set_Extended_enat,Y: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(X,Y) = bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(X),Y)) ) ).

% diff_shunt_var
tff(fact_6883_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).

% diff_shunt_var
tff(fact_6884_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% diff_shunt_var
tff(fact_6885_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% diff_shunt_var
tff(fact_6886_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu,B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( pp(B3)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,B3),zero_zero_nat)) ) ) )
         => ( ! [Uv: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv,Uw) )
               => ! [N2: nat] :
                    ( ( Xa = 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))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = 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,Xa),Mi2))
                             => ( Y = some_nat(Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),some_nat(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,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))),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,aa_num_num(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,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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,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,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_succ(Summary2,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,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_6887_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( Xa = 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) )
               => ( ( Xa = 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,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [Va: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( ( pp(B3)
                           => ( Y = some_nat(one_one_nat) ) )
                          & ( ~ pp(B3)
                           => ( ( 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,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2),Xa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = 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),Xa))
                               => ( Y = some_nat(Ma2) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( 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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))),some_nat(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,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one))))),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,aa_num_num(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,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,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa),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,aa_num_num(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,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,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,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_pred(Summary2,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,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_6888_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf(fFalse,B3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf(A4,fFalse) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [N2: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = vEBT_Leaf(A4,B3) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2),Xa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                     => ( ( Y = 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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2),Xa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                       => ( ( Y = 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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = 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,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( Y = 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,Xa),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y = 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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(bit0,one)))))),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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Xa)),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,v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t_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(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,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))))),aa_option_nat_nat(the_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_option_nat_nat(the_nat,vEBT_vebt_mint(Summary2))))))),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,aa_num_num(bit0,one)))))),Summary2)),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(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),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_6889_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu,B3) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,B3),zero_zero_nat)) ) ) )
         => ( ! [Uv: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv,Uw) )
               => ! [N2: nat] :
                    ( ( Xa = 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))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),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(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = 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,Xa),Mi2))
                             => ( Y = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_6890_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( 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,Xa),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,B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),Xa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = 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(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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,aa_num_num(bit0,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_6891_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_6892_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_6893_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B3) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y = vEBT_Leaf(A4,fTrue) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y = vEBT_Leaf(A4,B3) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),Xa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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 = 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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),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,Xa),Mi2)),Xa),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,Xa),Mi2)),Mi2),Xa)),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),Summary2)),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(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),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_6894_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( Xa = 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) )
               => ( ( Xa = 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,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [Va: nat] :
                      ( ( Xa = 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,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2,Ve2),Xa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),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(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = 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),Xa))
                               => ( Y = one_one_nat ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( 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,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,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_6895_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),Xa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = 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(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2,Summary2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(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,Xa),Mi2)),Mi2),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,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_6896_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,B3),Xa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),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,Xa),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),Xa)),one_one_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(bit0,aa_num_num(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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(bit0,one))))))),one_one_nat)))))))))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_6897_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,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(X),Xa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,Xa),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),Xa)),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),Xa)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),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,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,aa_num_num(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,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,aa_num_num(bit0,one)))))),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,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_6898_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2),Xa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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)
                        <=> ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(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),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_6899_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa))
               => ( ( ( Xa = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xa)) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz2),Xa)) )
             => ( ! [V3: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb,Vc2),Xa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = 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(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),Xa))
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_6900_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S3),Xa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_6901_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa))
               => ~ ( ( ( Xa = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S3),Xa))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_6902_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa))
               => ( ( ( Xa = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B3) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2,S3),Xa))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_6903_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B3),Xa))
               => ~ ( ( ( Xa = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B3) )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = 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(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),Xa))
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,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,aa_num_num(bit0,one)))))),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,aa_num_num(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,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,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_6904_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb) )
                 => ( ( pp(Y)
                    <=> ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb),Xa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),Xa)) ) )
               => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2),Xa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_6905_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X = 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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb),Xa))
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),Xa))
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2),Xa))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_6906_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( X = 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(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb),Xa))
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V3),TreeList2,Vc2),Xa))
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2,Vd2),Xa))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_6907_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_al,X)) ) ).

% monoseq_arctan_series
tff(fact_6908_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6909_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_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_6910_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_6911_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_6912_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_6913_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_6914_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_6915_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_am(Z)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6916_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_an,Z)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6917_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_ao(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_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,aa_num_num(bit0,one))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6918_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_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,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_6919_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ap,X)) ) ) ) ).

% ln_series
tff(fact_6920_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6921_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_6922_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_6923_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6924_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6925_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_aq,X)) ) ) ).

% arctan_series
tff(fact_6926_of__nat__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_ar(F),A3) ).

% of_nat_prod
tff(fact_6927_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_as(F)),A3) ).

% of_nat_prod
tff(fact_6928_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_at(F)),A3) ).

% of_nat_prod
tff(fact_6929_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = groups73079841787564623at_rat(aTP_Lamm_au(F),A3) ).

% of_nat_prod
tff(fact_6930_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = groups705719431365010083at_int(aTP_Lamm_av(F),A3) ).

% of_nat_prod
tff(fact_6931_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_aw(F)),A3) ).

% of_nat_prod
tff(fact_6932_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_6933_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_6934_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_6935_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_6936_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A3)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ax(F)),A3) ).

% of_int_prod
tff(fact_6937_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_complex(ring_17405671764205052669omplex,groups705719431365010083at_int(F,A3)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_ay(F)),A3) ).

% of_int_prod
tff(fact_6938_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A3)) = groups73079841787564623at_rat(aTP_Lamm_az(F),A3) ).

% of_int_prod
tff(fact_6939_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A3)) = groups705719431365010083at_int(aTP_Lamm_ba(F),A3) ).

% of_int_prod
tff(fact_6940_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A3)) = groups2316167850115554303t_real(aTP_Lamm_bb(F),A3) ).

% of_int_prod
tff(fact_6941_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups1705073143266064639nt_int(F,A3)) = groups7440179247065528705omplex(aTP_Lamm_bc(F),A3) ).

% of_int_prod
tff(fact_6942_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A3)) = groups1072433553688619179nt_rat(aTP_Lamm_bd(F),A3) ).

% of_int_prod
tff(fact_6943_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_be(F),A3) ).

% of_int_prod
tff(fact_6944_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6945_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_6946_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_6947_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_6948_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_6949_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_6950_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_6951_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_6952_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_6953_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_6954_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_6955_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_6956_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_6957_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_6958_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_6959_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6960_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_6961_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_bf(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6962_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_bg(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6963_prod_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))
       => ( aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6964_prod_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))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6965_prod_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))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6966_prod_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))
       => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6967_prod_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(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6968_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,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_6969_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,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_6970_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_6971_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_6972_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_6973_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6974_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6975_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6976_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_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(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6977_prod_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)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6978_prod_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)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,N))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6979_prod_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)))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6980_prod_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(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6981_prod_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))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6982_prod_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))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6983_prod_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))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6984_prod_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(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6985_prod_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(times_times_real,aa_set_nat_real(groups129246275422532515t_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(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6986_prod_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))
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6987_prod_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(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6988_prod_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(times_times_nat,aa_set_nat_nat(groups708209901874060359at_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(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6989_prod_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)))
     => ( aa_set_nat_real(groups129246275422532515t_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(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_real(groups129246275422532515t_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)))) ) ) ).

% prod.ub_add_nat
tff(fact_6990_prod_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)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups73079841787564623at_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)))) ) ) ).

% prod.ub_add_nat
tff(fact_6991_prod_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)))
     => ( groups705719431365010083at_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(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups705719431365010083at_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)))) ) ) ).

% prod.ub_add_nat
tff(fact_6992_prod_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(groups708209901874060359at_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(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups708209901874060359at_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)))) ) ) ).

% prod.ub_add_nat
tff(fact_6993_dbl__dec__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_dec_def
tff(fact_6994_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_6995_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_6996_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_6997_pochhammer__Suc__prod,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_bl(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6998_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_bm,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6999_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aTP_Lamm_bn(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7000_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aTP_Lamm_bo(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7001_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_bp,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_7002_pochhammer__prod__rev,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_set_nat_complex(groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_bq(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7003_pochhammer__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_br(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7004_pochhammer__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_bs(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7005_pochhammer__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_bt(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7006_pochhammer__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_bu(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)) ).

% pochhammer_prod_rev
tff(fact_7007_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_7008_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_7009_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(groups129246275422532515t_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_bv(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_7010_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups73079841787564623at_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_bw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_7011_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups705719431365010083at_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_bx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_7012_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_by(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% prod.in_pairs
tff(fact_7013_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_bq(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7014_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_br(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7015_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_bs(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7016_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_bt(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7017_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_bu(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_7018_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,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_7019_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_7020_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_7021_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_7022_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_7023_suminf__zero,axiom,
    suminf_real(aTP_Lamm_bz) = zero_zero_real ).

% suminf_zero
tff(fact_7024_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_ah) = zero_zero_nat ).

% suminf_zero
tff(fact_7025_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ca) = zero_zero_int ).

% suminf_zero
tff(fact_7026_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_cb) ).

% pi_series
tff(fact_7027_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_7028_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),one_one_real)) ) ).

% prod_le_1
tff(fact_7029_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),one_one_real)) ) ).

% prod_le_1
tff(fact_7030_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),one_one_real)) ) ).

% prod_le_1
tff(fact_7031_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7032_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7033_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7034_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A3)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7035_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7036_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),one_one_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),one_one_int)) ) ).

% prod_le_1
tff(fact_7037_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),one_one_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups705719431365010083at_int(F,A3)),one_one_int)) ) ).

% prod_le_1
tff(fact_7038_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_7039_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_7040_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_7041_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_7042_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_7043_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_7044_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_7045_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),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(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_7046_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_7047_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_7048_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N))) ).

% add_neg_numeral_special(5)
tff(fact_7049_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_7050_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_7051_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_7052_diff__numeral__special_I6_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_7053_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_7054_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_7055_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_7056_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_7057_diff__numeral__special_I5_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_7058_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N))) ).

% diff_numeral_special(5)
tff(fact_7059_int__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_ar(F),A3) ).

% int_prod
tff(fact_7060_int__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = groups705719431365010083at_int(aTP_Lamm_av(F),A3) ).

% int_prod
tff(fact_7061_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_7062_num__induct,axiom,
    ! [P: fun_num_bool,X: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X4: num] :
            ( pp(aa_num_bool(P,X4))
           => pp(aa_num_bool(P,aa_num_num(inc,X4))) )
       => pp(aa_num_bool(P,X)) ) ) ).

% num_induct
tff(fact_7063_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) ).

% add_inc
tff(fact_7064_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_7065_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_7066_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_7067_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_7068_inc_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit1,X)) = aa_num_num(bit0,aa_num_num(inc,X)) ).

% inc.simps(3)
tff(fact_7069_inc_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_7070_add__One,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),one) = aa_num_num(inc,X) ).

% add_One
tff(fact_7071_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,X),aa_num_num(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_7072_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_cc,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_7073_numeral__inc,axiom,
    ! [X: num] : numera6690914467698888265omplex(aa_num_num(inc,X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% numeral_inc
tff(fact_7074_numeral__inc,axiom,
    ! [X: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% numeral_inc
tff(fact_7075_numeral__inc,axiom,
    ! [X: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% numeral_inc
tff(fact_7076_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% numeral_inc
tff(fact_7077_numeral__inc,axiom,
    ! [X: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% numeral_inc
tff(fact_7078_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_7079_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) ).

% pi_ge_two
tff(fact_7080_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_7081_prod__int__plus__eq,axiom,
    ! [I: nat,J: 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),J))) = groups1705073143266064639nt_int(aTP_Lamm_cc,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),J)))) ).

% prod_int_plus_eq
tff(fact_7082_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7083_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_7084_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_le_two
tff(fact_7085_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_7086_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_ge_zero
tff(fact_7087_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_7088_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_7089_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_7090_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_7091_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_7092_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))) ).

% arctan_lbound
tff(fact_7093_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A3: set_nat,N: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A3)),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_cd(F),N),A3) ).

% prod_power_distrib
tff(fact_7094_prod__power__distrib,axiom,
    ! [F: fun_int_int,A3: set_int,N: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A3)),N) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_ce(F),N),A3) ).

% prod_power_distrib
tff(fact_7095_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),N) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cf(F),N)),A3) ).

% prod_power_distrib
tff(fact_7096_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_7097_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_7098_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I3)),aa_real_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(G,A3))) ) ).

% prod_mono
tff(fact_7099_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A3))
         => ( 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,aa_nat_real(F,I3)),aa_nat_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),aa_set_nat_real(groups129246275422532515t_real(G),A3))) ) ).

% prod_mono
tff(fact_7100_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I3)),aa_int_real(G,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(G,A3))) ) ).

% prod_mono
tff(fact_7101_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A3))
         => ( 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_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(G,A3))) ) ).

% prod_mono
tff(fact_7102_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A3))
         => ( 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_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(G,A3))) ) ).

% prod_mono
tff(fact_7103_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A3))
         => ( 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_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3))) ) ).

% prod_mono
tff(fact_7104_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A3)),groups4696554848551431203al_nat(G,A3))) ) ).

% prod_mono
tff(fact_7105_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,A3))) ) ).

% prod_mono
tff(fact_7106_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,I3)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(G,A3))) ) ).

% prod_mono
tff(fact_7107_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I3)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3))) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups705719431365010083at_int(F,A3)),groups705719431365010083at_int(G,A3))) ) ).

% prod_mono
tff(fact_7108_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A3))) ) ).

% prod_nonneg
tff(fact_7109_prod__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3))) ) ).

% prod_nonneg
tff(fact_7110_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3))) ) ).

% prod_nonneg
tff(fact_7111_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A3))) ) ).

% prod_pos
tff(fact_7112_prod__pos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3))) ) ).

% prod_pos
tff(fact_7113_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3))) ) ).

% prod_pos
tff(fact_7114_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A3))) ) ).

% prod_ge_1
tff(fact_7115_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A3))) ) ).

% prod_ge_1
tff(fact_7116_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A3))) ) ).

% prod_ge_1
tff(fact_7117_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A3))) ) ).

% prod_ge_1
tff(fact_7118_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A3))) ) ).

% prod_ge_1
tff(fact_7119_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3))) ) ).

% prod_ge_1
tff(fact_7120_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A3))) ) ).

% prod_ge_1
tff(fact_7121_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A3))) ) ).

% prod_ge_1
tff(fact_7122_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A3))) ) ).

% prod_ge_1
tff(fact_7123_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),groups705719431365010083at_int(F,A3))) ) ).

% prod_ge_1
tff(fact_7124_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_7125_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,aa_num_num(bit0,one))),N)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(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_7126_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(aa_num_num(bit0,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(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7127_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(aa_num_num(bit0,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7128_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(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7129_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),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7130_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),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7131_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),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7132_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_7133_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q3,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_7134_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_aq,X)) ) ).

% summable_arctan_series
tff(fact_7135_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_7136_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cg(I),F)) ).

% summable_single
tff(fact_7137_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ch(I),F)) ).

% summable_single
tff(fact_7138_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ci(I),F)) ).

% summable_single
tff(fact_7139_summable__zero,axiom,
    summable_real(aTP_Lamm_bz) ).

% summable_zero
tff(fact_7140_summable__zero,axiom,
    summable_nat(aTP_Lamm_ah) ).

% summable_zero
tff(fact_7141_summable__zero,axiom,
    summable_int(aTP_Lamm_ca) ).

% summable_zero
tff(fact_7142_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_7143_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7144_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ck(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7145_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cl(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7146_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_7147_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_7148_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_7149_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)))
    <=> unique6319869463603278526ux_int(unique5052692396658037445od_int(N,M)) ) ).

% dvd_numeral_simp
tff(fact_7150_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_7151_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_7152_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_7153_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_7154_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_7155_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_7156_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_7157_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_7158_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_7159_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(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_7160_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_7161_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_7162_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7163_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7164_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_7165_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_7166_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_7167_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7168_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_7169_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7170_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7171_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7172_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7173_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [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,N2))),aa_nat_real(G,N2))) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7174_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N4: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [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,N2))),aa_nat_real(G,N2))) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7175_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cm,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7176_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(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_co(F),Z)) ) ) ).

% powser_insidea
tff(fact_7177_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(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_cq(F),Z)) ) ) ).

% powser_insidea
tff(fact_7178_suminf__le,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(G,N2)))
     => ( summable_real(F)
       => ( summable_real(G)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G))) ) ) ) ).

% suminf_le
tff(fact_7179_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G))) ) ) ) ).

% suminf_le
tff(fact_7180_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
     => ( summable_int(F)
       => ( summable_int(G)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G))) ) ) ) ).

% suminf_le
tff(fact_7181_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_7182_sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real)) ).

% sin_le_one
tff(fact_7183_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),aa_real_real(abs_abs_real,X))) ).

% abs_sin_x_le_abs_x
tff(fact_7184_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7185_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7186_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7187_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7188_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_7189_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_7190_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_7191_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_7192_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_7193_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 )
        <=> ! [N3: nat] : aa_nat_real(F,N3) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7194_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 )
        <=> ! [N3: nat] : aa_nat_nat(F,N3) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7195_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 )
        <=> ! [N3: nat] : aa_nat_int(F,N3) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7196_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_7197_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_7198_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_7199_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_7200_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_7201_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_7202_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_bf(F)) ).

% summable_0_powser
tff(fact_7203_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_bg(F)) ).

% summable_0_powser
tff(fact_7204_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_bf(F)) ).

% summable_zero_power'
tff(fact_7205_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_bg(F)) ).

% summable_zero_power'
tff(fact_7206_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_cr(F)) ).

% summable_zero_power'
tff(fact_7207_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X))) ).

% sin_ge_minus_one
tff(fact_7208_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cs(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7209_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7210_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cs(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7211_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7212_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),one_one_real)) ).

% abs_sin_le_one
tff(fact_7213_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_cu(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7214_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_cv(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7215_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_cw(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7216_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_cx(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7217_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_cx(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_cx(F)))) ) ).

% summable_rabs
tff(fact_7218_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)))
        <=> ? [I4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_7219_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)))
        <=> ? [I4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_7220_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)))
        <=> ? [I4: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I4))) ) ) ) ).

% suminf_pos_iff
tff(fact_7221_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_7222_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_7223_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_7224_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(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_cn(F),Z)) ) ) ).

% powser_inside
tff(fact_7225_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(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_cp(F),Z)) ) ) ).

% powser_inside
tff(fact_7226_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_7227_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_7228_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_7229_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_7230_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_7231_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_bh(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_7232_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_7233_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7234_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(aa_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_7235_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_cy(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_cy(F)))) ) ).

% summable_norm
tff(fact_7236_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_cw(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_cw(F)))) ) ).

% summable_norm
tff(fact_7237_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),aa_num_real(numeral_numeral_real,aa_num_num(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_7238_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_7239_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(aa_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_7240_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_7241_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_7242_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(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_cs(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7243_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_cn(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_ct(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7244_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_cs(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7245_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cn(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_cn(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7246_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)
       => ? [N7: nat] :
          ! [N8: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N8))
           => 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_cz(F),N8)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7247_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)
       => ? [N7: nat] :
          ! [N8: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N8))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_da(F),N8)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7248_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_7249_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_cn(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7250_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7251_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M7: 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))),M7))
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_db(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_7252_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_7253_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_7254_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),aa_num_real(numeral_numeral_real,aa_num_num(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_7255_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,aa_num_real(numeral_numeral_real,aa_num_num(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_7256_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_7257_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7258_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7259_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7260_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_7261_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,aa_num_real(numeral_numeral_real,aa_num_num(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_7262_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)),aa_num_real(numeral_numeral_real,aa_num_num(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_7263_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7264_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7265_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))
       => ? [X4: 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(sin_real,X4) = 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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_7266_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,aa_num_num(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_7267_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,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7268_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,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_7269_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,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_7270_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,aa_num_num(bit0,N))) ) ) ) ).

% divmod_divmod_step
tff(fact_7271_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7272_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,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7273_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7274_minus__one__div__numeral,axiom,
    ! [N: num] : aa_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_7275_one__div__minus__numeral,axiom,
    ! [N: num] : aa_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_7276_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_dc(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_7277_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_dd(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_7278_pochhammer__code,axiom,
    ! [N: nat,A: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_de(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_7279_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_df(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_7280_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_dg(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_7281_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B,product_Pair_int_int(Q3,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B),product_Pair_int_int(Q3,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_7282_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_7283_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_7284_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_7285_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_7286_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( 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_int(Ys))))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_7287_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(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_7288_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_size_list_bool(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys),N) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(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_7289_product__nth,axiom,
    ! [N: nat,Xs: list_bool,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_size_list_bool(Xs)),size_size_list_nat(Ys))))
     => ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys),N) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(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_7290_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_int] :
      ( 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_bool(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_7291_product__nth,axiom,
    ! [N: nat,Xs: list_nat,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_nat(Xs)),size_size_list_num(Ys))))
     => ( nth_Pr8326237132889035090at_num(product_nat_num(Xs,Ys),N) = product_Pair_nat_num(aa_nat_nat(nth_nat(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_7292_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa_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_7293_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa_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_7294_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero_int,product_Pair_int_int(zero_zero_int,K)) ).

% eucl_rel_int_by0
tff(fact_7295_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => eucl_rel_int(K,L,product_Pair_int_int(Q3,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_7296_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7297_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = Acc ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_7298_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q3,R2))
     => ( ( B != zero_zero_int )
       => 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,Q3)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q3)),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(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_7299_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q3,R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),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))
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_7300_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q3,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B),product_Pair_int_int(Q3,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_7301_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_7302_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,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7303_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T2) )
               => ( Y != aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7304_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7305_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,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = 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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = 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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_7306_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_7307_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_7308_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_7309_bit_Ocompl__eq__compl__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(bit_ri7919022796975470100ot_int,X) = aa_int_int(bit_ri7919022796975470100ot_int,Y) )
    <=> ( X = Y ) ) ).

% bit.compl_eq_compl_iff
tff(fact_7310_bit_Odouble__compl,axiom,
    ! [X: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_ri7919022796975470100ot_int,X)) = X ).

% bit.double_compl
tff(fact_7311_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7312_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_7313_bit_Oxor__compl__right,axiom,
    ! [X: int,Y: int] : aa_int_int(bit_se6526347334894502574or_int(X),aa_int_int(bit_ri7919022796975470100ot_int,Y)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)) ).

% bit.xor_compl_right
tff(fact_7314_bit_Oxor__compl__left,axiom,
    ! [X: int,Y: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_ri7919022796975470100ot_int,X)),Y) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)) ).

% bit.xor_compl_left
tff(fact_7315_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_7316_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7317_in__set__replicate,axiom,
    ! [X: option_nat,N: nat,Y: option_nat] :
      ( member_option_nat(X,set_option_nat2(replicate_option_nat(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7318_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_7319_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( member_set_nat(X,set_set_nat2(replicate_set_nat(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7320_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_7321_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_7322_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7323_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7324_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ? [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N,A))))
          & pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7325_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7326_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7327_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N,A))))
         => pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7328_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7329_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_7330_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_7331_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_7332_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_7333_bit_Ocompl__one,axiom,
    bit_ri7632146776885996613nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_7334_bit_Ocompl__one,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% bit.compl_one
tff(fact_7335_bit_Ocompl__zero,axiom,
    bit_ri7632146776885996613nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_7336_bit_Ocompl__zero,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.compl_zero
tff(fact_7337_bit_Odisj__cancel__right,axiom,
    ! [X: code_integer] : bit_se1080825931792720795nteger(X,bit_ri7632146776885996613nteger(X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_cancel_right
tff(fact_7338_bit_Odisj__cancel__right,axiom,
    ! [X: int] : aa_int_int(bit_se1409905431419307370or_int(X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_cancel_right
tff(fact_7339_bit_Odisj__cancel__left,axiom,
    ! [X: code_integer] : bit_se1080825931792720795nteger(bit_ri7632146776885996613nteger(X),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_cancel_left
tff(fact_7340_bit_Odisj__cancel__left,axiom,
    ! [X: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_cancel_left
tff(fact_7341_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(X,bit_ri7632146776885996613nteger(X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_right
tff(fact_7342_bit_Oxor__cancel__right,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_right
tff(fact_7343_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(bit_ri7632146776885996613nteger(X),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_left
tff(fact_7344_bit_Oxor__cancel__left,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_left
tff(fact_7345_bit_Oxor__one__right,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(X) ).

% bit.xor_one_right
tff(fact_7346_bit_Oxor__one__right,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_right
tff(fact_7347_bit_Oxor__one__left,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),X) = bit_ri7632146776885996613nteger(X) ).

% bit.xor_one_left
tff(fact_7348_bit_Oxor__one__left,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_left
tff(fact_7349_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_7350_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_7351_minus__not__numeral__eq,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_ri7632146776885996613nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(inc,N)) ).

% minus_not_numeral_eq
tff(fact_7352_minus__not__numeral__eq,axiom,
    ! [N: num] : aa_int_int(uminus_uminus_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(inc,N)) ).

% minus_not_numeral_eq
tff(fact_7353_even__not__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri7632146776885996613nteger(A)))
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).

% even_not_iff
tff(fact_7354_even__not__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,A)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_not_iff
tff(fact_7355_set__replicate,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] :
      ( ( N != zero_zero_nat )
     => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ).

% set_replicate
tff(fact_7356_set__replicate,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( N != zero_zero_nat )
     => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ).

% set_replicate
tff(fact_7357_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_7358_set__replicate,axiom,
    ! [N: nat,X: extended_enat] :
      ( ( N != zero_zero_nat )
     => ( set_Extended_enat2(replic7216382294607269926d_enat(N,X)) = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ) ).

% set_replicate
tff(fact_7359_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_7360_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_7361_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_7362_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_7363_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,aa_num_real(numeral_numeral_real,N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_7364_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_7365_not__one__eq,axiom,
    bit_ri7632146776885996613nteger(one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_7366_not__one__eq,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_7367_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7368_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7369_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_7370_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_7371_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_7372_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_7373_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_7374_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,aa_num_num(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,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7375_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,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7376_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,aa_num_num(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(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7377_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,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7378_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_7379_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_7380_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7381_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).

% cos_int_2pin
tff(fact_7382_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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7383_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(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_7384_of__int__not__eq,axiom,
    ! [K: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,K)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(ring_1_of_int_int,K)) ).

% of_int_not_eq
tff(fact_7385_take__bit__not__take__bit,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se2923211474154528505it_int(N),A))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_ri7919022796975470100ot_int,A)) ).

% take_bit_not_take_bit
tff(fact_7386_take__bit__not__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_ri7919022796975470100ot_int,B)) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(bit_se2923211474154528505it_int(N),B) ) ) ).

% take_bit_not_iff
tff(fact_7387_of__int__not__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_not_numeral
tff(fact_7388_not__diff__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B) ).

% not_diff_distrib
tff(fact_7389_not__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B) ).

% not_add_distrib
tff(fact_7390_cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real)) ).

% cos_le_one
tff(fact_7391_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_7392_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_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_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_7400_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_7401_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_7402_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_7403_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_7404_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = plus_p5714425477246183910nteger(bit_ri7632146776885996613nteger(A),one_one_Code_integer) ).

% minus_eq_not_plus_1
tff(fact_7405_minus__eq__not__plus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).

% minus_eq_not_plus_1
tff(fact_7406_minus__eq__not__minus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = bit_ri7632146776885996613nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)) ).

% minus_eq_not_minus_1
tff(fact_7407_minus__eq__not__minus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ).

% minus_eq_not_minus_1
tff(fact_7408_not__eq__complement,axiom,
    ! [A: code_integer] : bit_ri7632146776885996613nteger(A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),one_one_Code_integer) ).

% not_eq_complement
tff(fact_7409_not__eq__complement,axiom,
    ! [A: int] : aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ).

% not_eq_complement
tff(fact_7410_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_7411_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_7412_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_7413_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X))) ).

% cos_ge_minus_one
tff(fact_7414_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_7415_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X))),one_one_real)) ).

% abs_cos_le_one
tff(fact_7416_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_7417_minus__numeral__inc__eq,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,N))) = bit_ri7632146776885996613nteger(numera6620942414471956472nteger(N)) ).

% minus_numeral_inc_eq
tff(fact_7418_minus__numeral__inc__eq,axiom,
    ! [N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)) ).

% minus_numeral_inc_eq
tff(fact_7419_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_7420_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_7421_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(divide_divide_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_7422_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,aa_num_num(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,aa_num_num(bit0,one))),K)) ) ).

% even_not_iff_int
tff(fact_7423_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7424_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_7425_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_7426_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(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).

% tan_half
tff(fact_7427_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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_7428_not__numeral__Bit0__eq,axiom,
    ! [N: num] : bit_ri7632146776885996613nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,N))) ).

% not_numeral_Bit0_eq
tff(fact_7429_not__numeral__Bit0__eq,axiom,
    ! [N: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% not_numeral_Bit0_eq
tff(fact_7430_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,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(2)
tff(fact_7431_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(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_7432_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_7433_set__replicate__conv__if,axiom,
    ! [N: nat,X: produc3843707927480180839at_nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = bot_bo228742789529271731at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Pr3765526544606949372at_nat(replic2264142908078655527at_nat(N,X)) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_7434_set__replicate__conv__if,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = bot_bo2099793752762293965at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,X)) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_7435_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_7436_set__replicate__conv__if,axiom,
    ! [N: nat,X: extended_enat] :
      ( ( ( N = zero_zero_nat )
       => ( set_Extended_enat2(replic7216382294607269926d_enat(N,X)) = bot_bo7653980558646680370d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( set_Extended_enat2(replic7216382294607269926d_enat(N,X)) = insert_Extended_enat(X,bot_bo7653980558646680370d_enat) ) ) ) ).

% set_replicate_conv_if
tff(fact_7437_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_7438_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_7439_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_7440_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_7441_sin__double,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,X))),aa_complex_complex(cos_complex,X)) ).

% sin_double
tff(fact_7442_sin__double,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,X))),aa_real_real(cos_real,X)) ).

% sin_double
tff(fact_7443_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_7444_cos__is__zero,axiom,
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X4) = 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
tff(fact_7445_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_7446_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(aa_num_num(bit0,one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(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,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7447_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7448_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,aa_num_num(bit0,N))) ).

% or_not_numerals(3)
tff(fact_7449_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_7450_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_7451_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))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),pi))
            & ( aa_real_real(cos_real,X4) = 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 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_7452_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_7453_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7454_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_7455_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_7456_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_7457_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7458_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_7459_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(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7460_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,aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7461_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(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7462_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7463_sin__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7464_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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7465_cos__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7466_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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7467_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),aa_num_real(numeral_numeral_real,aa_num_num(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_7468_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X4))) ) ) ).

% lemma_tan_total
tff(fact_7469_tan__total,axiom,
    ! [Y: real] :
    ? [X4: 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
tff(fact_7470_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7471_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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_7472_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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_7473_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_7474_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7475_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_7476_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_7477_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),aa_num_real(numeral_numeral_real,aa_num_num(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_7478_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_7479_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_7480_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_7481_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X3: 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,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7482_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7483_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_7484_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(aa_num_num(bit0,aa_num_num(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_7485_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_7486_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(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7487_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,aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7488_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(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7489_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,aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7490_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(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7491_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,aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7492_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(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7493_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7494_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(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7495_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7496_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(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7497_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7498_cos__double,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7499_cos__double,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7500_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) ).

% cos_sin_eq
tff(fact_7501_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(aa_num_num(bit0,one)))),X)) ).

% cos_sin_eq
tff(fact_7502_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) ).

% sin_cos_eq
tff(fact_7503_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(aa_num_num(bit0,one)))),X)) ).

% sin_cos_eq
tff(fact_7504_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_7505_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),aa_num_real(numeral_numeral_real,aa_num_num(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_7506_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)),aa_num_real(numeral_numeral_real,aa_num_num(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_7507_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7508_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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_7509_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(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_7510_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_7511_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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_7512_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,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_7513_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(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_7514_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_7515_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_7516_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_7517_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_7518_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X3: 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,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X3: 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,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7519_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7520_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7521_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_7522_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(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_7523_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))
     => ? [Z2: 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_7524_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,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7525_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),pi))
            & ( X = aa_real_real(cos_real,T2) )
            & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_7526_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,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7527_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)),aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7528_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7529_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,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = 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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = 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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_7530_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,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7531_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N3: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N3: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7532_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)),aa_num_real(numeral_numeral_real,aa_num_num(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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7533_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
              & ( X = aa_real_real(cos_real,T2) )
              & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7534_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
          & ( X = aa_real_real(cos_real,T2) )
          & ( Y = aa_real_real(sin_real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7535_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T2)),aa_real_real(sin_real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7536_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A3: set_nat,B4: 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(A3),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B4))
         => ( ( A3 = B4 )
           => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A3),X),X_1)) ) ) ) ) ).

% obtain_set_succ
tff(fact_7537_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A3: 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(A3),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
         => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A3),X),X_1)) ) ) ) ).

% obtain_set_pred
tff(fact_7538_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,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_7539_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_complex(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 )
           => ( aa_set_nat_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 )
           => ( aa_set_nat_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_7540_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)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7541_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_real(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 )
           => ( aa_set_nat_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 )
           => ( aa_set_nat_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_7542_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_7543_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))
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),A)) ) ) ) ).

% pred_none_empty
tff(fact_7544_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))
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X5)) ) ) ) ).

% succ_none_empty
tff(fact_7545_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_7546_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ah),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7547_sum_Oneutral__const,axiom,
    ! [A3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_dh,A3) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7548_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bz),A3) = zero_zero_real ).

% sum.neutral_const
tff(fact_7549_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_ai,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_7550_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A3: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A3)) = groups7754918857620584856omplex(aTP_Lamm_di(F),A3) ).

% of_nat_sum
tff(fact_7551_of__nat__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_ar(F),A3) ).

% of_nat_sum
tff(fact_7552_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_as(F)),A3) ).

% of_nat_sum
tff(fact_7553_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups2906978787729119204at_rat(aTP_Lamm_au(F),A3) ).

% of_nat_sum
tff(fact_7554_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups3539618377306564664at_int(aTP_Lamm_av(F),A3) ).

% of_nat_sum
tff(fact_7555_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aw(F)),A3) ).

% of_nat_sum
tff(fact_7556_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_at(F)),A3) ).

% of_nat_sum
tff(fact_7557_of__int__sum,axiom,
    ! [F: fun_complex_int,A3: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A3)) = groups7754918857620584856omplex(aTP_Lamm_dj(F),A3) ).

% of_int_sum
tff(fact_7558_of__int__sum,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A3)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ax(F)),A3) ).

% of_int_sum
tff(fact_7559_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A3)) = groups8778361861064173332t_real(aTP_Lamm_bb(F),A3) ).

% of_int_sum
tff(fact_7560_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups4538972089207619220nt_int(F,A3)) = groups3049146728041665814omplex(aTP_Lamm_bc(F),A3) ).

% of_int_sum
tff(fact_7561_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A3)) = groups3906332499630173760nt_rat(aTP_Lamm_bd(F),A3) ).

% of_int_sum
tff(fact_7562_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_be(F),A3) ).

% of_int_sum
tff(fact_7563_sum_Oempty,axiom,
    ! [G: fun_Ex1451620480498224165t_real] : groups4148127829035722712t_real(G,bot_bo7653980558646680370d_enat) = zero_zero_real ).

% sum.empty
tff(fact_7564_sum_Oempty,axiom,
    ! [G: fun_Ex4152069839811072081at_rat] : groups1392844769737527556at_rat(G,bot_bo7653980558646680370d_enat) = zero_zero_rat ).

% sum.empty
tff(fact_7565_sum_Oempty,axiom,
    ! [G: fun_Ex7660298398204468809at_nat] : groups2027974829824023292at_nat(G,bot_bo7653980558646680370d_enat) = zero_zero_nat ).

% sum.empty
tff(fact_7566_sum_Oempty,axiom,
    ! [G: fun_Ex3482447378695272101at_int] : groups2025484359314973016at_int(G,bot_bo7653980558646680370d_enat) = zero_zero_int ).

% sum.empty
tff(fact_7567_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_7568_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_7569_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_7570_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_7571_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_7572_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_7573_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7574_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7575_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7576_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( groups2906978787729119204at_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7577_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7578_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7579_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7580_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups4541462559716669496nt_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7581_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5693394587270226106ex_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7582_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups2027974829824023292at_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7583_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),F3))
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7584_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),F3))
           => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7585_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( finite6177210948735845034at_nat(F3)
     => ( ( groups977919841031483927at_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X3,F3)
           => ( aa_Pro4243173770044397973at_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7586_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( ( groups2027974829824023292at_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),F3))
           => ( aa_Extended_enat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7587_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),F3))
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7588_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A3) = zero_zero_real )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & ( aa_nat_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7589_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups2316167850115554303t_real(F,A3) = zero_zero_real )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7590_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups766887009212190081x_real(F,A3) = zero_zero_real )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7591_prod__zero__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( groups97031904164794029t_real(F,A3) = zero_zero_real )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & ( aa_Ext2541572767299617110t_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7592_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( groups73079841787564623at_rat(F,A3) = zero_zero_rat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7593_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups1072433553688619179nt_rat(F,A3) = zero_zero_rat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7594_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups225925009352817453ex_rat(F,A3) = zero_zero_rat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7595_prod__zero__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( groups2245840878043517529at_rat(F,A3) = zero_zero_rat )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & ( aa_Extended_enat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7596_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups1707563613775114915nt_nat(F,A3) = zero_zero_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7597_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups861055069439313189ex_nat(F,A3) = zero_zero_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7598_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_7599_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_7600_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_dk(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_dk(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7601_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_dl(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_dl(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7602_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_dm(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_dm(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7603_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_dn(A),B),S2) = aa_Ext2541572767299617110t_real(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_dn(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_7604_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_do(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_do(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7605_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dp(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dp(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7606_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: 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(aTP_Lamm_dq(A),B),S2) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_dq(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7607_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_dr(A),B),S2) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_dr(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7608_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_ds(A),B),S2) = aa_Extended_enat_rat(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_ds(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_7609_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_dt(A),B),S2) = aa_real_nat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_dt(A),B),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta'
tff(fact_7610_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_du(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_du(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7611_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_dv(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_dv(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7612_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_dw(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_dw(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7613_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_dx(A),B),S2) = aa_Ext2541572767299617110t_real(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_dx(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_7614_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_dy(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dy(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7615_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dz(A),B),S2) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dz(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7616_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: 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(aTP_Lamm_ea(A),B),S2) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ea(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7617_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_eb(A),B),S2) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_eb(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7618_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_ec(A),B),S2) = aa_Extended_enat_rat(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S2))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_ec(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_7619_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ed(A),B),S2) = aa_real_nat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ed(A),B),S2) = zero_zero_nat ) ) ) ) ).

% sum.delta
tff(fact_7620_sum__abs,axiom,
    ! [F: fun_nat_real,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cx(F)),A3))) ).

% sum_abs
tff(fact_7621_sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A3))),groups4538972089207619220nt_int(aTP_Lamm_ee(F),A3))) ).

% sum_abs
tff(fact_7622_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ef(A3),F)) ) ).

% summable_If_finite_set
tff(fact_7623_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_eg(A3),F)) ) ).

% summable_If_finite_set
tff(fact_7624_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_eh(A3),F)) ) ).

% summable_If_finite_set
tff(fact_7625_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_ei(P),F)) ) ).

% summable_If_finite
tff(fact_7626_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_ej(P),F)) ) ).

% summable_If_finite
tff(fact_7627_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_ek(P),F)) ) ).

% summable_If_finite
tff(fact_7628_prod__pos__nat__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A3)))
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7629_prod__pos__nat__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A3)))
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7630_prod__pos__nat__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( finite6177210948735845034at_nat(A3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4077766827762148844at_nat(F,A3)))
      <=> ! [X3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X3,A3)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7631_prod__pos__nat__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups2880970938130013265at_nat(F,A3)))
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7632_prod__pos__nat__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)))
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7633_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cx(F)),A3))) ).

% sum_abs_ge_zero
tff(fact_7634_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A3: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_ee(F),A3))) ).

% sum_abs_ge_zero
tff(fact_7635_sin__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))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_7636_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))) = aa_rat_rat(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_7637_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(aa_int_fun_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_7638_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_7639_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))
       => ( aa_set_nat_real(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))
       => ( aa_set_nat_real(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_set_nat_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_7640_sum__zero__power,axiom,
    ! [A3: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_bf(C)),A3) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_bf(C)),A3) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_7641_sum__zero__power,axiom,
    ! [A3: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_el(C),A3) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_el(C),A3) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_7642_sum__zero__power,axiom,
    ! [A3: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bg(C)),A3) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bg(C)),A3) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_7643_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_7644_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: fun_nat_complex,D3: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_em(C),D3)),A3) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D3,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_em(C),D3)),A3) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_7645_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: fun_nat_rat,D3: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_en(C),D3),A3) = aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D3,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_en(C),D3),A3) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_7646_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: fun_nat_real,D3: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A3))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(C),D3)),A3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D3,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A3))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(C),D3)),A3) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_7647_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_7648_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_7649_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_7650_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) = aa_set_nat_real(groups6591440286371151544t_real(F),N4) ) ) ) ).

% suminf_finite
tff(fact_7651_sum__norm__le,axiom,
    ! [S2: set_option_nat,F: fun_op5246294523634652747omplex,G: fun_option_nat_real] :
      ( ! [X4: option_nat] :
          ( member_option_nat(X4,S2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_opt3066276567374697100omplex(F,X4))),aa_option_nat_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3209896439523537802omplex(F,S2))),groups4518532050878116744t_real(G,S2))) ) ).

% sum_norm_le
tff(fact_7652_sum__norm__le,axiom,
    ! [S2: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X4))),aa_real_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S2))),groups8097168146408367636l_real(G,S2))) ) ).

% sum_norm_le
tff(fact_7653_sum__norm__le,axiom,
    ! [S2: set_set_nat,F: fun_set_nat_complex,G: fun_set_nat_real] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,S2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X4))),aa_set_nat_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S2))),groups5107569545109728110t_real(G,S2))) ) ).

% sum_norm_le
tff(fact_7654_sum__norm__le,axiom,
    ! [S2: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X4))),aa_int_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S2))),groups8778361861064173332t_real(G,S2))) ) ).

% sum_norm_le
tff(fact_7655_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X4))),aa_nat_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S2))),aa_set_nat_real(groups6591440286371151544t_real(G),S2))) ) ).

% sum_norm_le
tff(fact_7656_sum__norm__le,axiom,
    ! [S2: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X4))),aa_complex_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S2))),groups5808333547571424918x_real(G,S2))) ) ).

% sum_norm_le
tff(fact_7657_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X4))),aa_nat_real(G,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S2))),aa_set_nat_real(groups6591440286371151544t_real(G),S2))) ) ).

% sum_norm_le
tff(fact_7658_norm__sum,axiom,
    ! [F: fun_nat_complex,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cw(F)),A3))) ).

% norm_sum
tff(fact_7659_norm__sum,axiom,
    ! [F: fun_complex_complex,A3: set_complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A3))),groups5808333547571424918x_real(aTP_Lamm_ep(F),A3))) ).

% norm_sum
tff(fact_7660_norm__sum,axiom,
    ! [F: fun_nat_real,A3: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cy(F)),A3))) ).

% norm_sum
tff(fact_7661_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa: real] :
      ( ( aa_real_complex(complex2(X),Y) = aa_real_complex(real_V4546457046886955230omplex,Xa) )
    <=> ( ( X = Xa )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_7662_complex__of__real__code,axiom,
    ! [X5: real] : aa_real_complex(real_V4546457046886955230omplex,X5) = aa_real_complex(complex2(X5),zero_zero_real) ).

% complex_of_real_code
tff(fact_7663_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_7664_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_7665_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_7666_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T4: 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,T3))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_real_real(I,aa_real_real(J,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(J,A4)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B3)),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 ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_real_real(H,aa_real_real(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7667_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_real,H: fun_int_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_int_real(I,aa_real_int(J,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(J,A4)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B3)),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 ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_int_real(H,aa_real_int(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7668_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_complex_real(I,aa_real_complex(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A4)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I,B3)),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 ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_complex_real(H,aa_real_complex(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7669_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_Extended_enat,S2: set_real,I: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T4: set_Extended_enat,G: fun_real_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_Ext2541572767299617110t_real(I,aa_rea1194391308913012860d_enat(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A4)),minus_925952699566721837d_enat(T4,T3))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_Ext2541572767299617110t_real(I,B3)),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 ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T3))
                       => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_Ext2541572767299617110t_real(H,aa_rea1194391308913012860d_enat(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups4148127829035722712t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7670_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T4: set_real,G: fun_int_real,H: fun_real_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_real_int(I,aa_int_real(J,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(J,A4)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_real_real(H,aa_int_real(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7671_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T4: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_int_int(I,aa_int_int(J,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(J,A4)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_int_real(H,aa_int_int(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7672_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T4: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_complex_int(I,aa_int_complex(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A4)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_complex_real(H,aa_int_complex(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7673_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_Extended_enat,S2: set_int,I: fun_Ex3482447378695272101at_int,J: fun_in5914218760682845459d_enat,T4: set_Extended_enat,G: fun_int_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_Extended_enat_int(I,aa_int_Extended_enat(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_int_Extended_enat(J,A4)),minus_925952699566721837d_enat(T4,T3))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                 => ( aa_int_Extended_enat(J,aa_Extended_enat_int(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_Extended_enat_int(I,B3)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T3))
                       => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_Ext2541572767299617110t_real(H,aa_int_Extended_enat(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups4148127829035722712t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7674_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T4: set_real,G: fun_complex_real,H: fun_real_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A4: complex] :
              ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(S2,S4)))
             => ( aa_real_complex(I,aa_complex_real(J,A4)) = A4 ) )
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_complex_real(J,A4)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S2,S4))) )
               => ( ! [A4: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A4),S4))
                     => ( aa_complex_real(G,A4) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A4),S2))
                         => ( aa_real_real(H,aa_complex_real(J,A4)) = aa_complex_real(G,A4) ) )
                     => ( groups5808333547571424918x_real(G,S2) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7675_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T4: set_int,G: fun_complex_real,H: fun_int_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A4: complex] :
              ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(S2,S4)))
             => ( aa_int_complex(I,aa_complex_int(J,A4)) = A4 ) )
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A4)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S2,S4))) )
               => ( ! [A4: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A4),S4))
                     => ( aa_complex_real(G,A4) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A4: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A4),S2))
                         => ( aa_int_real(H,aa_complex_int(J,A4)) = aa_complex_real(G,A4) ) )
                     => ( groups5808333547571424918x_real(G,S2) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7676_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_eq(G)))) = groups8097168146408367636l_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7677_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_er(G)))) = groups8778361861064173332t_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7678_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_es(G)))) = groups5808333547571424918x_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7679_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A3,collec4429806609662206161d_enat(aTP_Lamm_et(G)))) = groups4148127829035722712t_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7680_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_eu(G)))) = groups1300246762558778688al_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7681_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ev(G)))) = groups3906332499630173760nt_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7682_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ew(G)))) = groups5058264527183730370ex_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7683_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G,minus_925952699566721837d_enat(A3,collec4429806609662206161d_enat(aTP_Lamm_ex(G)))) = groups1392844769737527556at_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7684_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_ey(G)))) = groups1935376822645274424al_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7685_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ez(G)))) = groups4541462559716669496nt_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7686_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7687_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7688_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7689_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7690_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7691_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X4)),aa_Extended_enat_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7692_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7693_sum__strict__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7694_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7695_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3))) ) ) ) ).

% sum_strict_mono
tff(fact_7696_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(A3),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_fb(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7697_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fc(A3),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_fd(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7698_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_fe(A3),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_ff(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7699_sum_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex1451620480498224165t_real,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups4148127829035722712t_real(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_fg(A3),P))) = groups4148127829035722712t_real(aa_fun1090353380788292083t_real(aTP_Lamm_fh(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7700_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(A3),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_fi(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7701_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fj(A3),P))) = groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_fk(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7702_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_rat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fc(A3),P))) = groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_fl(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7703_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_rat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_fe(A3),P))) = groups5058264527183730370ex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_fm(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7704_sum_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( groups1392844769737527556at_rat(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_fg(A3),P))) = groups1392844769737527556at_rat(aa_fun6658670135733114143at_rat(aTP_Lamm_fn(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7705_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_nat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( groups1935376822645274424al_nat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(A3),P))) = groups1935376822645274424al_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_fo(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7706_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7707_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7708_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7709_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3))) )
       => ( ( groups4148127829035722712t_real(F,S) = zero_zero_real )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => ( aa_Ext2541572767299617110t_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7710_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_7711_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_7712_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_7713_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(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 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7714_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I3))) )
       => ( ( groups1392844769737527556at_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => ( aa_Extended_enat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7715_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7716_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B4: real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))) )
       => ( ( groups8097168146408367636l_real(F,S) = B4 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7717_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B4: real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))) )
       => ( ( groups8778361861064173332t_real(F,S) = B4 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7718_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( ( groups5808333547571424918x_real(F,S) = B4 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7719_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real,B4: real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3))) )
       => ( ( groups4148127829035722712t_real(F,S) = B4 )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7720_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B4: 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) = B4 )
         => ( 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)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7721_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B4: 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) = B4 )
         => ( 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)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7722_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B4: 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) = B4 )
         => ( 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)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7723_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(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) = B4 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7724_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,B4: rat,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I3))) )
       => ( ( groups1392844769737527556at_rat(F,S) = B4 )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7725_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B4: nat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
       => ( ( groups1935376822645274424al_nat(F,S) = B4 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I)),B4)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7726_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),T))
                    & ( aa_int_int(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7727_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa2),T))
                    & ( aa_complex_int(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_complex_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7728_sum__le__included,axiom,
    ! [S: set_int,T: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: fun_Ex3482447378695272101at_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa2: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),T))
                    & ( aa_Extended_enat_int(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_Ext2541572767299617110t_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups4148127829035722712t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7729_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),T))
                    & ( aa_int_complex(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_int_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7730_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa2),T))
                    & ( aa_complex_complex(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7731_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: fun_Ex3741802288643809319omplex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa2: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),T))
                    & ( aa_Ext8748583412541189208omplex(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_Ext2541572767299617110t_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups4148127829035722712t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7732_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_int,G: fun_int_real,I: fun_in5914218760682845459d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),T))
                    & ( aa_int_Extended_enat(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_int_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7733_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_complex,G: fun_complex_real,I: fun_co6496365337113880337d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
               => ? [Xa2: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa2),T))
                    & ( aa_com3683005661447327866d_enat(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_complex_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7734_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: fun_Ex5852502975177633731d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G,X4))) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
               => ? [Xa2: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),T))
                    & ( aa_Ext7969882492699152700d_enat(I,Xa2) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,Xa2))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S)),groups4148127829035722712t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_7735_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))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa2),T))
                    & ( aa_nat_nat(I,Xa2) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa2))) ) )
           => 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_7736_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A3) = zero_zero_real )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A3))
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7737_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A3) = zero_zero_real )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A3))
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7738_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A3) = zero_zero_real )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A3))
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7739_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
       => ( ( groups4148127829035722712t_real(F,A3) = zero_zero_real )
        <=> ! [X3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
             => ( aa_Ext2541572767299617110t_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7740_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( ( groups1300246762558778688al_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A3))
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7741_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( ( groups2906978787729119204at_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A3))
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7742_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( ( groups3906332499630173760nt_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A3))
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7743_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( ( groups5058264527183730370ex_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A3))
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7744_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X4))) )
       => ( ( groups1392844769737527556at_rat(F,A3) = zero_zero_rat )
        <=> ! [X3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
             => ( aa_Extended_enat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7745_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( ( groups1935376822645274424al_nat(F,A3) = zero_zero_nat )
        <=> ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),A3))
             => ( aa_real_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7746_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7747_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7748_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7749_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7750_sum__mono__inv,axiom,
    ! [F: fun_Ex4152069839811072081at_rat,I5: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,I: extended_enat] :
      ( ( groups1392844769737527556at_rat(F,I5) = groups1392844769737527556at_rat(G,I5) )
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I3)),aa_Extended_enat_rat(G,I3))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
           => ( aa_Extended_enat_rat(F,I) = aa_Extended_enat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7751_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7752_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7753_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7754_sum__mono__inv,axiom,
    ! [F: fun_Ex7660298398204468809at_nat,I5: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,I: extended_enat] :
      ( ( groups2027974829824023292at_nat(F,I5) = groups2027974829824023292at_nat(G,I5) )
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,I3)),aa_Extended_enat_nat(G,I3))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
           => ( aa_Extended_enat_nat(F,I) = aa_Extended_enat_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7755_sum__mono__inv,axiom,
    ! [F: fun_real_int,I5: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I5) = groups1932886352136224148al_int(G,I5) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_7756_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))
     => ( ! [X1: real,Y1: real,X22: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,X1),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7757_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))
     => ( ! [X1: real,Y1: real,X22: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,X1),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7758_sum_Orelated,axiom,
    ! [R3: fun_re1255401033344146673l_bool,S2: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X22: real,Y22: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,X1),X22))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,Y1),Y22)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_Ext2541572767299617110t_real(H,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,groups4148127829035722712t_real(H,S2)),groups4148127829035722712t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7759_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))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X22),Y22))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7760_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))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X22),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7761_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))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X22),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_complex_rat(H,X4)),aa_complex_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7762_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_bool,S2: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X22: rat,Y22: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X1),X22))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y22)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X22),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_Extended_enat_rat(H,X4)),aa_Extended_enat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups1392844769737527556at_rat(H,S2)),groups1392844769737527556at_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7763_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))
     => ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X1),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y22))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7764_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))
     => ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X1),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y22))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_complex_nat(H,X4)),aa_complex_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7765_sum_Orelated,axiom,
    ! [R3: fun_nat_fun_nat_bool,S2: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X1),X22))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y1),Y22)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y22))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_Extended_enat_nat(H,X4)),aa_Extended_enat_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,groups2027974829824023292at_nat(H,S2)),groups2027974829824023292at_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_7766_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups8097168146408367636l_real(G,A3) != zero_zero_real )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A3))
           => ( aa_real_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7767_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups8778361861064173332t_real(G,A3) != zero_zero_real )
     => ~ ! [A4: int] :
            ( pp(aa_set_int_bool(member_int(A4),A3))
           => ( aa_int_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7768_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A3: set_real] :
      ( ( groups1300246762558778688al_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A3))
           => ( aa_real_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7769_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( pp(aa_set_nat_bool(member_nat(A4),A3))
           => ( aa_nat_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7770_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: int] :
            ( pp(aa_set_int_bool(member_int(A4),A3))
           => ( aa_int_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7771_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A3: set_real] :
      ( ( groups1935376822645274424al_nat(G,A3) != zero_zero_nat )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A3))
           => ( aa_real_nat(G,A4) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7772_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A3: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A3) != zero_zero_nat )
     => ~ ! [A4: int] :
            ( pp(aa_set_int_bool(member_int(A4),A3))
           => ( aa_int_nat(G,A4) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7773_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_int,A3: set_real] :
      ( ( groups1932886352136224148al_int(G,A3) != zero_zero_int )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A3))
           => ( aa_real_int(G,A4) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7774_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A3: set_nat] :
      ( ( groups3539618377306564664at_int(G,A3) != zero_zero_int )
     => ~ ! [A4: nat] :
            ( pp(aa_set_nat_bool(member_nat(A4),A3))
           => ( aa_nat_int(G,A4) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7775_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_nat,A3: set_nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) != zero_zero_nat )
     => ~ ! [A4: nat] :
            ( pp(aa_set_nat_bool(member_nat(A4),A3))
           => ( aa_nat_nat(G,A4) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7776_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_7777_sum_Oneutral,axiom,
    ! [A3: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A3))
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A3) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_7778_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A3) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_7779_sum_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => ( aa_int_int(G,X4) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_7780_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X5)),aa_int_real(G,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7781_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X5)),aa_complex_real(G,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7782_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
       => ( ? [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X5)),aa_Ext2541572767299617110t_real(G,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7783_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
       => ( ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X5)),aa_nat_rat(G,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7784_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
       => ( ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X5)),aa_int_rat(G,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7785_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X5)),aa_complex_rat(G,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7786_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,X4)),aa_Extended_enat_rat(G,X4))) )
       => ( ? [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X5)),aa_Extended_enat_rat(G,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7787_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X5)),aa_int_nat(G,X5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7788_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X5)),aa_complex_nat(G,X5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7789_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G,X4))) )
       => ( ? [X5: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X5)),aa_Extended_enat_nat(G,X5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2027974829824023292at_nat(F,A3)),groups2027974829824023292at_nat(G,A3))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7790_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_7791_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% bounded_nat_set_is_finite
tff(fact_7792_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_7793_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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_pos2
tff(fact_7794_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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_pos2
tff(fact_7795_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_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_pos2
tff(fact_7796_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_7797_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_7798_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_7799_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_7800_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(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] :
                ( pp(aa_set_complex_bool(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_7801_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,I)))
         => ( ! [I3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1392844769737527556at_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_7802_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_7803_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(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_7804_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_7805_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_7806_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_7807_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(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_7808_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1392844769737527556at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_7809_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_7810_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_7811_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_7812_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_7813_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5))) ) ).

% sum_mono
tff(fact_7814_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5))) ) ).

% sum_mono
tff(fact_7815_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5))) ) ).

% sum_mono
tff(fact_7816_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).

% sum_mono
tff(fact_7817_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).

% sum_mono
tff(fact_7818_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5))) ) ).

% sum_mono
tff(fact_7819_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5))) ) ).

% sum_mono
tff(fact_7820_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I3)),aa_nat_nat(G,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K5))) ) ).

% sum_mono
tff(fact_7821_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),K5)),aa_set_nat_real(groups6591440286371151544t_real(G),K5))) ) ).

% sum_mono
tff(fact_7822_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,I3)),aa_int_int(G,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4538972089207619220nt_int(F,K5)),groups4538972089207619220nt_int(G,K5))) ) ).

% sum_mono
tff(fact_7823_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T4,S2)))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T4) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7824_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,T4) = groups8778361861064173332t_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7825_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T4) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7826_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,T4) = groups4148127829035722712t_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7827_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T4,S2)))
             => ( aa_real_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,T4) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7828_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups3906332499630173760nt_rat(G,T4) = groups3906332499630173760nt_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7829_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,T4) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7830_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Extended_enat_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_rat(G,X4) = aa_Extended_enat_rat(H,X4) ) )
           => ( groups1392844769737527556at_rat(G,T4) = groups1392844769737527556at_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7831_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T4,S2)))
             => ( aa_real_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,T4) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7832_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,T4) = groups4541462559716669496nt_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7833_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T4,S2)))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7834_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7835_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7836_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Ext2541572767299617110t_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,S2) = groups4148127829035722712t_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7837_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T4,S2)))
             => ( aa_real_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7838_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7839_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S2))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7840_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Extended_enat_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
               => ( aa_Extended_enat_rat(G,X4) = aa_Extended_enat_rat(H,X4) ) )
           => ( groups1392844769737527556at_rat(G,S2) = groups1392844769737527556at_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7841_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T4))
       => ( ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(T4,S2)))
             => ( aa_real_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),S2))
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7842_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S2))
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7843_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T4) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7844_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T4) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7845_sum_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,T4) = groups4148127829035722712t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7846_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T4) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7847_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T4) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7848_sum_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Extended_enat_rat(G,X4) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G,T4) = groups1392844769737527556at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7849_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T4) = groups4541462559716669496nt_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7850_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T4) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7851_sum_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,T4) = groups2027974829824023292at_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7852_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T4) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7853_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7854_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7855_sum_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,S2) = groups4148127829035722712t_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7856_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7857_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7858_sum_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Extended_enat_rat(G,X4) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G,S2) = groups1392844769737527556at_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7859_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S2)))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7860_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7861_sum_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S2),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S2)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,S2) = groups2027974829824023292at_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7862_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S2)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7863_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A3)))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A3) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7864_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A3)))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B4)))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G,A3) = groups8778361861064173332t_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7865_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A3)))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A3) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7866_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B4: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B4),C4))
         => ( ! [A4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A4),minus_925952699566721837d_enat(C4,A3)))
               => ( aa_Ext2541572767299617110t_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B4)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) )
               => ( groups4148127829035722712t_real(G,A3) = groups4148127829035722712t_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7867_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A3)))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A3) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7868_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A3)))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B4)))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) )
               => ( groups3906332499630173760nt_rat(G,A3) = groups3906332499630173760nt_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7869_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A3)))
               => ( aa_complex_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A3) = groups5058264527183730370ex_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7870_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B4: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B4),C4))
         => ( ! [A4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A4),minus_925952699566721837d_enat(C4,A3)))
               => ( aa_Extended_enat_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B4)))
                 => ( aa_Extended_enat_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1392844769737527556at_rat(G,C4) = groups1392844769737527556at_rat(H,C4) )
               => ( groups1392844769737527556at_rat(G,A3) = groups1392844769737527556at_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7871_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A3)))
               => ( aa_real_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A3) = groups1935376822645274424al_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7872_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B4: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A3)))
               => ( aa_int_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B4)))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) )
               => ( groups4541462559716669496nt_nat(G,A3) = groups4541462559716669496nt_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7873_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A3)))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A3) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7874_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A3)))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B4)))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A3) = groups8778361861064173332t_real(H,B4) )
              <=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7875_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A3)))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A3) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7876_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B4: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B4),C4))
         => ( ! [A4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A4),minus_925952699566721837d_enat(C4,A3)))
               => ( aa_Ext2541572767299617110t_real(G,A4) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B4)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,A3) = groups4148127829035722712t_real(H,B4) )
              <=> ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7877_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A3)))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A3) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7878_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A3)))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B4)))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A3) = groups3906332499630173760nt_rat(H,B4) )
              <=> ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7879_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B4),C4))
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A3)))
               => ( aa_complex_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B4)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A3) = groups5058264527183730370ex_rat(H,B4) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7880_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B4: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B4),C4))
         => ( ! [A4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A4),minus_925952699566721837d_enat(C4,A3)))
               => ( aa_Extended_enat_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B4)))
                 => ( aa_Extended_enat_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1392844769737527556at_rat(G,A3) = groups1392844769737527556at_rat(H,B4) )
              <=> ( groups1392844769737527556at_rat(G,C4) = groups1392844769737527556at_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7881_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B4),C4))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A3)))
               => ( aa_real_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B4)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A3) = groups1935376822645274424al_nat(H,B4) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7882_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B4: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B4),C4))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A3)))
               => ( aa_int_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B4)))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,A3) = groups4541462559716669496nt_nat(H,B4) )
              <=> ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7883_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_fp(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_7884_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fq(F),U)))) ) ).

% finite_less_ub
tff(fact_7885_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7886_sum__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7887_sum__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7888_sum__mono2,axiom,
    ! [B4: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B4))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7889_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7890_sum__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7891_sum__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7892_sum__mono2,axiom,
    ! [B4: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B4))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7893_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7894_sum__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B4))) ) ) ) ).

% sum_mono2
tff(fact_7895_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A3))) ) ).

% sum_nonneg
tff(fact_7896_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A3))) ) ).

% sum_nonneg
tff(fact_7897_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_7898_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_7899_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3))) ) ).

% sum_nonneg
tff(fact_7900_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A3))) ) ).

% sum_nonneg
tff(fact_7901_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3))) ) ).

% sum_nonneg
tff(fact_7902_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A3))) ) ).

% sum_nonneg
tff(fact_7903_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups3539618377306564664at_int(F,A3))) ) ).

% sum_nonneg
tff(fact_7904_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3))) ) ).

% sum_nonneg
tff(fact_7905_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_7906_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_7907_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7908_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7909_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_7910_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_7911_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_7912_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A3)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_7913_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X4)),zero_zero_int)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,A3)),zero_zero_int)) ) ).

% sum_nonpos
tff(fact_7914_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_7915_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_7916_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_7917_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,aa_set_nat_real(groups6591440286371151544t_real(F),I5)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_7918_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A3))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_7919_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A3))
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A3) = aa_set_nat_real(groups6591440286371151544t_real(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_7920_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_7921_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_7922_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_7923_infinite__growing,axiom,
    ! [X7: set_Extended_enat] :
      ( ( X7 != bot_bo7653980558646680370d_enat )
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),X7))
           => ? [Xa2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),X7))
                & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X4),Xa2)) ) )
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X7)) ) ) ).

% infinite_growing
tff(fact_7924_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),X7))
           => ? [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),X7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Xa2)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X7)) ) ) ).

% infinite_growing
tff(fact_7925_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),X7))
           => ? [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),X7))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Xa2)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X7)) ) ) ).

% infinite_growing
tff(fact_7926_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),X7))
           => ? [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),X7))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Xa2)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X7)) ) ) ).

% infinite_growing
tff(fact_7927_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),X7))
           => ? [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),X7))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa2)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X7)) ) ) ).

% infinite_growing
tff(fact_7928_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),X7))
           => ? [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),X7))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Xa2)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X7)) ) ) ).

% infinite_growing
tff(fact_7929_ex__min__if__finite,axiom,
    ! [S2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S2))
            & ~ ? [Xa2: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),S2))
                  & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7930_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),S2))
            & ~ ? [Xa2: real] :
                  ( pp(aa_set_real_bool(member_real(Xa2),S2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7931_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S2))
     => ( ( S2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),S2))
            & ~ ? [Xa2: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa2),S2))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7932_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S2))
     => ( ( S2 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),S2))
            & ~ ? [Xa2: num] :
                  ( pp(aa_set_num_bool(member_num(Xa2),S2))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7933_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),S2))
            & ~ ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),S2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7934_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),S2))
            & ~ ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),S2))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7935_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
            & ( aa_nat_real(F,X5) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7936_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & ( aa_int_real(F,X5) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7937_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
            & ( aa_complex_real(F,X5) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7938_prod__zero,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
            & ( aa_Ext2541572767299617110t_real(F,X5) = zero_zero_real ) )
       => ( groups97031904164794029t_real(F,A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7939_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A3))
            & ( aa_nat_rat(F,X5) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7940_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & ( aa_int_rat(F,X5) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7941_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
            & ( aa_complex_rat(F,X5) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7942_prod__zero,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ? [X5: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A3))
            & ( aa_Extended_enat_rat(F,X5) = zero_zero_rat ) )
       => ( groups2245840878043517529at_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7943_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A3))
            & ( aa_int_nat(F,X5) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7944_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A3))
            & ( aa_complex_nat(F,X5) = zero_zero_nat ) )
       => ( groups861055069439313189ex_nat(F,A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7945_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_7946_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_7947_finite__lists__length__le,axiom,
    ! [A3: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_fr(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7948_finite__lists__length__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( finite6177210948735845034at_nat(A3)
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_fs(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7949_finite__lists__length__le,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_ft(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7950_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_fu(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7951_finite__lists__length__le,axiom,
    ! [A3: set_bool,N: nat] :
      ( finite_finite_bool(A3)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fv(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7952_finite__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_fw(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7953_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fx(A3),N))) ) ).

% finite_lists_length_le
tff(fact_7954_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_7955_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_7956_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_7957_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,B: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7958_sum__strict__mono2,axiom,
    ! [B4: set_int,A3: set_int,B: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B4,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,B)))
           => ( ! [X4: int] :
                  ( pp(aa_set_int_bool(member_int(X4),B4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7959_sum__strict__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,B: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B4,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7960_sum__strict__mono2,axiom,
    ! [B4: set_Extended_enat,A3: set_Extended_enat,B: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B4))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(B4,A3)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A3)),groups4148127829035722712t_real(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7961_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,B: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7962_sum__strict__mono2,axiom,
    ! [B4: set_int,A3: set_int,B: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B4,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,B)))
           => ( ! [X4: int] :
                  ( pp(aa_set_int_bool(member_int(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7963_sum__strict__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,B: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B4,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7964_sum__strict__mono2,axiom,
    ! [B4: set_Extended_enat,A3: set_Extended_enat,B: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B4))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(B4,A3)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,B)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A3)),groups1392844769737527556at_rat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7965_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,B: real,F: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B4,A3)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B)))
           => ( ! [X4: real] :
                  ( pp(aa_set_real_bool(member_real(X4),B4))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7966_sum__strict__mono2,axiom,
    ! [B4: set_int,A3: set_int,B: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B4,A3)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,B)))
           => ( ! [X4: int] :
                  ( pp(aa_set_int_bool(member_int(X4),B4))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B4))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7967_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I),A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7968_member__le__sum,axiom,
    ! [I: extended_enat,A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(A3,insert_Extended_enat(I,bot_bo7653980558646680370d_enat))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),groups4148127829035722712t_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7969_member__le__sum,axiom,
    ! [I: real,A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I),A3))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7970_member__le__sum,axiom,
    ! [I: int,A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I),A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A3,insert_int(I,bot_bot_set_int))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7971_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I),A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A3,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,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7972_member__le__sum,axiom,
    ! [I: extended_enat,A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),A3))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(A3,insert_Extended_enat(I,bot_bo7653980558646680370d_enat))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X4))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I)),groups1392844769737527556at_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7973_member__le__sum,axiom,
    ! [I: real,A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I),A3))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(A3,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,X4))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7974_member__le__sum,axiom,
    ! [I: int,A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I),A3))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(A3,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,X4))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7975_member__le__sum,axiom,
    ! [I: nat,A3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(member_nat(I),A3))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(A3,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,X4))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7976_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(member_complex(I),A3))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I)),groups5693394587270226106ex_nat(F,A3))) ) ) ) ).

% member_le_sum
tff(fact_7977_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_fy(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fy(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_fz(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7978_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_ga(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ga(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_gb(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7979_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_gc(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_gc(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_gd(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7980_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ge(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ge(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_gf(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7981_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex1451620480498224165t_real,Y: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_gg(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_gg(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_gh(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7982_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_gi(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_gi(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_gj(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7983_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_gk(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_gk(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_gl(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7984_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_rat,Y: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_gm(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_gm(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_gn(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7985_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_rat,Y: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_go(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_go(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aa_fun1496239077807485399x_bool(aTP_Lamm_gp(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7986_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex4152069839811072081at_rat,Y: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun5314330785533781769t_bool(aTP_Lamm_gq(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun5314330785533781769t_bool(aTP_Lamm_gq(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun5314330785533781769t_bool(aa_fun4769541129632704515t_bool(aTP_Lamm_gr(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7987_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_7988_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,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_7989_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7990_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: fun_se2015444840911768724t_bool,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X4: extended_enat,S5: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S5))
             => ( ! [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,Y4)),aa_Extended_enat_rat(F,X4))) )
               => ( pp(aa_set3347977610044113549t_bool(P,S5))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,S5))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7991_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X4))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,insert_real(X4,S5))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7992_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S5))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S5))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S5))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7993_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S5))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X4))) )
               => ( pp(aa_set_int_bool(P,S5))
                 => pp(aa_set_int_bool(P,insert_int(X4,S5))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7994_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X4: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X4))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X4,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7995_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: fun_se2015444840911768724t_bool,F: fun_Ex8368542335153306515at_num] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S2))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [X4: extended_enat,S5: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S5))
             => ( ! [Y4: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Extended_enat_num(F,Y4)),aa_Extended_enat_num(F,X4))) )
               => ( pp(aa_set3347977610044113549t_bool(P,S5))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(X4,S5))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7996_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_bool,F: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X4: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X4))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,insert_real(X4,S5))) ) ) )
         => pp(aa_set_real_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7997_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X4: nat,S5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S5))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,Y4)),aa_nat_num(F,X4))) )
               => ( pp(aa_set_nat_bool(P,S5))
                 => pp(aa_set_nat_bool(P,insert_nat(X4,S5))) ) ) )
         => pp(aa_set_nat_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7998_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_bool,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X4: int,S5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S5))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,Y4)),aa_int_num(F,X4))) )
               => ( pp(aa_set_int_bool(P,S5))
                 => pp(aa_set_int_bool(P,insert_int(X4,S5))) ) ) )
         => pp(aa_set_int_bool(P,S2)) ) ) ) ).

% finite_ranking_induct
tff(fact_7999_finite__linorder__min__induct,axiom,
    ! [A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [B3: extended_enat,A6: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A6))
             => ( ! [X5: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A6))
                   => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B3),X5)) )
               => ( pp(aa_set3347977610044113549t_bool(P,A6))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(B3,A6))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8000_finite__linorder__min__induct,axiom,
    ! [A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( ! [X5: real] :
                    ( pp(aa_set_real_bool(member_real(X5),A6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),X5)) )
               => ( pp(aa_set_real_bool(P,A6))
                 => pp(aa_set_real_bool(P,insert_real(B3,A6))) ) ) )
         => pp(aa_set_real_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8001_finite__linorder__min__induct,axiom,
    ! [A3: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A6: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A6))
             => ( ! [X5: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X5),A6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),X5)) )
               => ( pp(aa_set_rat_bool(P,A6))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A6))) ) ) )
         => pp(aa_set_rat_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8002_finite__linorder__min__induct,axiom,
    ! [A3: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A6: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A6))
             => ( ! [X5: num] :
                    ( pp(aa_set_num_bool(member_num(X5),A6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B3),X5)) )
               => ( pp(aa_set_num_bool(P,A6))
                 => pp(aa_set_num_bool(P,insert_num(B3,A6))) ) ) )
         => pp(aa_set_num_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8003_finite__linorder__min__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( ! [X5: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X5),A6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),X5)) )
               => ( pp(aa_set_nat_bool(P,A6))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A6))) ) ) )
         => pp(aa_set_nat_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8004_finite__linorder__min__induct,axiom,
    ! [A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( ! [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),A6))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B3),X5)) )
               => ( pp(aa_set_int_bool(P,A6))
                 => pp(aa_set_int_bool(P,insert_int(B3,A6))) ) ) )
         => pp(aa_set_int_bool(P,A3)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8005_finite__linorder__max__induct,axiom,
    ! [A3: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(P,bot_bo7653980558646680370d_enat))
       => ( ! [B3: extended_enat,A6: set_Extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A6))
             => ( ! [X5: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X5),A6))
                   => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X5),B3)) )
               => ( pp(aa_set3347977610044113549t_bool(P,A6))
                 => pp(aa_set3347977610044113549t_bool(P,insert_Extended_enat(B3,A6))) ) ) )
         => pp(aa_set3347977610044113549t_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8006_finite__linorder__max__induct,axiom,
    ! [A3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B3: real,A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( ! [X5: real] :
                    ( pp(aa_set_real_bool(member_real(X5),A6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B3)) )
               => ( pp(aa_set_real_bool(P,A6))
                 => pp(aa_set_real_bool(P,insert_real(B3,A6))) ) ) )
         => pp(aa_set_real_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8007_finite__linorder__max__induct,axiom,
    ! [A3: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B3: rat,A6: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A6))
             => ( ! [X5: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X5),A6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),B3)) )
               => ( pp(aa_set_rat_bool(P,A6))
                 => pp(aa_set_rat_bool(P,insert_rat(B3,A6))) ) ) )
         => pp(aa_set_rat_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8008_finite__linorder__max__induct,axiom,
    ! [A3: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B3: num,A6: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A6))
             => ( ! [X5: num] :
                    ( pp(aa_set_num_bool(member_num(X5),A6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),B3)) )
               => ( pp(aa_set_num_bool(P,A6))
                 => pp(aa_set_num_bool(P,insert_num(B3,A6))) ) ) )
         => pp(aa_set_num_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8009_finite__linorder__max__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B3: nat,A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( ! [X5: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X5),A6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),B3)) )
               => ( pp(aa_set_nat_bool(P,A6))
                 => pp(aa_set_nat_bool(P,insert_nat(B3,A6))) ) ) )
         => pp(aa_set_nat_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8010_finite__linorder__max__induct,axiom,
    ! [A3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B3: int,A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( ! [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),A6))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),B3)) )
               => ( pp(aa_set_int_bool(P,A6))
                 => pp(aa_set_int_bool(P,insert_int(B3,A6))) ) ) )
         => pp(aa_set_int_bool(P,A3)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8011_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_gs(X),M)),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),I5)) ).

% sum_power_add
tff(fact_8012_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_gt(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_8013_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_gu(X),M),I5) = aa_int_int(aa_int_fun_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_8014_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gv(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)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),I5)) ).

% sum_power_add
tff(fact_8015_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_gw,M)))) ) ).

% finite_divisors_nat
tff(fact_8016_arcsin__minus,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(arcsin,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_8017_arcsin__le__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_eq_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_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_8018_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_8019_arcsin__eq__iff,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))
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_8020_arcsin__le__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_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_8021_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_8022_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_8023_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_8024_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8025_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))) = aa_rat_rat(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_8026_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(aa_int_fun_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_8027_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_8028_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(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,aa_set_nat_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_8029_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)) = aa_rat_rat(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_8030_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(aa_int_fun_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_8031_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_8032_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))
     => ( aa_set_nat_real(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)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8033_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))) = aa_rat_rat(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_8034_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(aa_int_fun_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_8035_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_8036_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)))
     => ( aa_set_nat_real(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))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8037_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8038_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8039_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8040_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8041_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups97031904164794029t_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8042_less__1__prod2,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,one_one_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,one_one_rat),aa_real_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8043_less__1__prod2,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,one_one_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,one_one_rat),aa_nat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8044_less__1__prod2,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,one_one_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,one_one_rat),aa_int_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8045_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8046_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_Extended_enat_rat(F,I)))
         => ( ! [I3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_Extended_enat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups2245840878043517529at_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_8047_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8048_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups97031904164794029t_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8049_less__1__prod,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,one_one_real),aa_real_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8050_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( 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_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5))) ) ) ) ).

% less_1_prod
tff(fact_8051_less__1__prod,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,one_one_real),aa_int_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8052_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8053_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_Extended_enat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups2245840878043517529at_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8054_less__1__prod,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,one_one_rat),aa_real_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8055_less__1__prod,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,one_one_rat),aa_nat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8056_less__1__prod,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,one_one_rat),aa_int_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_8057_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))
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8058_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8059_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_bk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8060_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,aa_set_nat_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)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8061_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_gx(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8062_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_gy(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8063_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)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gz(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_8064_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(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8065_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_8066_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(aTP_Lamm_ha(N)))) ) ).

% finite_roots_unity
tff(fact_8067_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_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_hb(N)))) ) ).

% finite_roots_unity
tff(fact_8068_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_hc(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8069_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_hd(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8070_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_he(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8071_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_hf(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8072_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_8073_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_8074_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))) = aa_rat_rat(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_8075_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(aa_int_fun_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_8076_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_8077_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)))
     => ( aa_set_nat_real(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,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_real(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_8078_prod__mono__strict,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,I3)),aa_complex_real(G,I3))) ) )
       => ( ( A3 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8079_prod__mono__strict,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),A3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,I3)),aa_Ext2541572767299617110t_real(G,I3))) ) )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups97031904164794029t_real(F,A3)),groups97031904164794029t_real(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8080_prod__mono__strict,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,I3)),aa_real_real(G,I3))) ) )
       => ( ( A3 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8081_prod__mono__strict,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A3))
           => ( 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_real,aa_nat_real(F,I3)),aa_nat_real(G,I3))) ) )
       => ( ( A3 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),aa_set_nat_real(groups129246275422532515t_real(G),A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8082_prod__mono__strict,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,I3)),aa_int_real(G,I3))) ) )
       => ( ( A3 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8083_prod__mono__strict,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A3))
           => ( 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,aa_complex_rat(F,I3)),aa_complex_rat(G,I3))) ) )
       => ( ( A3 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8084_prod__mono__strict,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),A3))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,I3)),aa_Extended_enat_rat(G,I3))) ) )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2245840878043517529at_rat(F,A3)),groups2245840878043517529at_rat(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8085_prod__mono__strict,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A3))
           => ( 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,aa_real_rat(F,I3)),aa_real_rat(G,I3))) ) )
       => ( ( A3 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8086_prod__mono__strict,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A3))
           => ( 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,aa_nat_rat(F,I3)),aa_nat_rat(G,I3))) ) )
       => ( ( A3 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8087_prod__mono__strict,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A3))
           => ( 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,aa_int_rat(F,I3)),aa_int_rat(G,I3))) ) )
       => ( ( A3 != bot_bot_set_int )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3))) ) ) ) ).

% prod_mono_strict
tff(fact_8088_even__prod__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3455450783089532116nteger(F,A3)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8089_even__prod__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3827104343326376752nteger(F,A3)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8090_even__prod__iff,axiom,
    ! [A3: set_complex,F: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A3)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8091_even__prod__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex3703362190075743318nteger] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups845124408420398302nteger(F,A3)))
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Ext3588433912111482887nteger(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8092_even__prod__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups1707563613775114915nt_nat(F,A3)))
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8093_even__prod__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A3)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8094_even__prod__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups2880970938130013265at_nat(F,A3)))
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_Extended_enat_nat(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8095_even__prod__iff,axiom,
    ! [A3: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A3)))
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8096_even__prod__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex3482447378695272101at_int] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups2878480467620962989at_int(F,A3)))
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Extended_enat_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8097_even__prod__iff,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A3)))
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(F,X3))) ) ) ) ).

% even_prod_iff
tff(fact_8098_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_hg(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8099_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_hh(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8100_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_hi(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8101_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I3))) )
     => ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => 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(A,I3)),B))),Delta)) )
         => 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,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hj(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8102_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I3))) )
     => ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => 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(A,I3)),B))),Delta)) )
         => 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,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_hk(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8103_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,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hl(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8104_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,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hm(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8105_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,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hn(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8106_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_int,A: fun_real_int,B: int,Delta: int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(X,I3))) )
     => ( ( groups1932886352136224148al_int(X,I5) = one_one_int )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(A,I3)),B))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_ho(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8107_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I3))) )
     => ( ( groups3539618377306564664at_int(X,I5) = one_one_int )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(A,I3)),B))),Delta)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_hp(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8108_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_8109_prod__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,B3))) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),A3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,A4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8110_prod__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,B3))) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),A3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,A4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8111_prod__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,B3))) )
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),A3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8112_prod__mono2,axiom,
    ! [B4: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B4))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B4,A3)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,B3))) )
         => ( ! [A4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A4),A3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,A4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups97031904164794029t_real(F,A3)),groups97031904164794029t_real(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8113_prod__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3))) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),A3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8114_prod__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A3),B4))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B3))) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),A3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8115_prod__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3))) )
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),A3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8116_prod__mono2,axiom,
    ! [B4: set_Extended_enat,A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A3),B4))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B4,A3)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_Extended_enat_rat(F,B3))) )
         => ( ! [A4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A4),A3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,A4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2245840878043517529at_rat(F,A3)),groups2245840878043517529at_rat(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8117_prod__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A3),B4))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B4,A3)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,B3))) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),A3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,A4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8118_prod__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A3),B4))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B4,A3)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_complex_int(F,B3))) )
         => ( ! [A4: complex] :
                ( pp(aa_set_complex_bool(member_complex(A4),A3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A4))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(F,B4))) ) ) ) ) ).

% prod_mono2
tff(fact_8119_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_hq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(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_hq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_8120_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_hr(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_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_hr(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_8121_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))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hs(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))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hs(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_8122_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_ht(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_8123_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_hu(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_8124_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))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hv(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_8125_prod__diff1,axiom,
    ! [A3: set_complex,F: fun_complex_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A3))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A3,insert_complex(A,bot_bot_set_complex))) = aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A3)),aa_complex_complex(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A3))
           => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A3,insert_complex(A,bot_bot_set_complex))) = groups3708469109370488835omplex(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8126_prod__diff1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex3741802288643809319omplex,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( aa_Ext8748583412541189208omplex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
           => ( groups4622424608036095791omplex(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_complex_complex(divide1717551699836669952omplex(groups4622424608036095791omplex(F,A3)),aa_Ext8748583412541189208omplex(F,A)) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
           => ( groups4622424608036095791omplex(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups4622424608036095791omplex(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8127_prod__diff1,axiom,
    ! [A3: set_real,F: fun_real_complex,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_real_bool(member_real(A),A3))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A3,insert_real(A,bot_bot_set_real))) = aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A3)),aa_real_complex(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A3))
           => ( groups713298508707869441omplex(F,minus_minus_set_real(A3,insert_real(A,bot_bot_set_real))) = groups713298508707869441omplex(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8128_prod__diff1,axiom,
    ! [A3: set_int,F: fun_int_complex,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_int_bool(member_int(A),A3))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A3,insert_int(A,bot_bot_set_int))) = aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A3)),aa_int_complex(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A3))
           => ( groups7440179247065528705omplex(F,minus_minus_set_int(A3,insert_int(A,bot_bot_set_int))) = groups7440179247065528705omplex(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8129_prod__diff1,axiom,
    ! [A3: set_nat,F: fun_nat_complex,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A3))
           => ( aa_set_nat_complex(groups6464643781859351333omplex(F),minus_minus_set_nat(A3,insert_nat(A,bot_bot_set_nat))) = aa_complex_complex(divide1717551699836669952omplex(aa_set_nat_complex(groups6464643781859351333omplex(F),A3)),aa_nat_complex(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A3))
           => ( aa_set_nat_complex(groups6464643781859351333omplex(F),minus_minus_set_nat(A3,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_complex(groups6464643781859351333omplex(F),A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8130_prod__diff1,axiom,
    ! [A3: set_complex,F: fun_complex_real,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( aa_complex_real(F,A) != zero_zero_real )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A3))
           => ( groups766887009212190081x_real(F,minus_811609699411566653omplex(A3,insert_complex(A,bot_bot_set_complex))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups766887009212190081x_real(F,A3)),aa_complex_real(F,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A3))
           => ( groups766887009212190081x_real(F,minus_811609699411566653omplex(A3,insert_complex(A,bot_bot_set_complex))) = groups766887009212190081x_real(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8131_prod__diff1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( aa_Ext2541572767299617110t_real(F,A) != zero_zero_real )
       => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
           => ( groups97031904164794029t_real(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups97031904164794029t_real(F,A3)),aa_Ext2541572767299617110t_real(F,A)) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
           => ( groups97031904164794029t_real(F,minus_925952699566721837d_enat(A3,insert_Extended_enat(A,bot_bo7653980558646680370d_enat))) = groups97031904164794029t_real(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8132_prod__diff1,axiom,
    ! [A3: set_real,F: fun_real_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( aa_real_real(F,A) != zero_zero_real )
       => ( ( pp(aa_set_real_bool(member_real(A),A3))
           => ( groups1681761925125756287l_real(F,minus_minus_set_real(A3,insert_real(A,bot_bot_set_real))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups1681761925125756287l_real(F,A3)),aa_real_real(F,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A3))
           => ( groups1681761925125756287l_real(F,minus_minus_set_real(A3,insert_real(A,bot_bot_set_real))) = groups1681761925125756287l_real(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8133_prod__diff1,axiom,
    ! [A3: set_int,F: fun_int_real,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( aa_int_real(F,A) != zero_zero_real )
       => ( ( pp(aa_set_int_bool(member_int(A),A3))
           => ( groups2316167850115554303t_real(F,minus_minus_set_int(A3,insert_int(A,bot_bot_set_int))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups2316167850115554303t_real(F,A3)),aa_int_real(F,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A3))
           => ( groups2316167850115554303t_real(F,minus_minus_set_int(A3,insert_int(A,bot_bot_set_int))) = groups2316167850115554303t_real(F,A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8134_prod__diff1,axiom,
    ! [A3: set_nat,F: fun_nat_real,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_nat_real(F,A) != zero_zero_real )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A3))
           => ( aa_set_nat_real(groups129246275422532515t_real(F),minus_minus_set_nat(A3,insert_nat(A,bot_bot_set_nat))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),aa_nat_real(F,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A3))
           => ( aa_set_nat_real(groups129246275422532515t_real(F),minus_minus_set_nat(A3,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_real(groups129246275422532515t_real(F),A3) ) ) ) ) ) ).

% prod_diff1
tff(fact_8135_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N7: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),M2))
                 => ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N8)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8136_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N7: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),M2))
                 => ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N8)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8137_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,N))) ).

% mask_eq_sum_exp
tff(fact_8138_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,aa_num_num(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,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,N))) ).

% mask_eq_sum_exp
tff(fact_8139_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)),aa_set_nat_complex(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_8140_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(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_rat_rat(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_8141_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(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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(aa_int_fun_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_8142_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)),aa_set_nat_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(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_8143_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_hx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8144_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_hy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8145_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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8146_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : aa_set_nat_real(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,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ia(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8147_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_8148_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,aa_num_num(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,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,N))) ).

% mask_eq_sum_exp_nat
tff(fact_8149_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ib),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,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8150_double__arith__series,axiom,
    ! [A: complex,D3: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ic(A),D3)),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(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D3))) ).

% double_arith_series
tff(fact_8151_double__arith__series,axiom,
    ! [A: rat,D3: rat,N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_id(A),D3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D3))) ).

% double_arith_series
tff(fact_8152_double__arith__series,axiom,
    ! [A: int,D3: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ie(A),D3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D3))) ).

% double_arith_series
tff(fact_8153_double__arith__series,axiom,
    ! [A: nat,D3: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_if(A),D3)),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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D3))) ).

% double_arith_series
tff(fact_8154_double__arith__series,axiom,
    ! [A: real,D3: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ig(A),D3)),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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D3))) ).

% double_arith_series
tff(fact_8155_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(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_8156_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(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)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8157_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_8158_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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_8159_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(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_8160_arith__series__nat,axiom,
    ! [A: nat,D3: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ih(A),D3)),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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D3)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8161_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ib),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,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8162_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8163_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_bounded
tff(fact_8164_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arcsin_ubound
tff(fact_8165_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_8166_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_8167_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(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_8168_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(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)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8169_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8170_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,aa_num_num(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_8171_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(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_8172_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8173_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,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8174_arith__series,axiom,
    ! [A: int,D3: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ie(A),D3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D3)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8175_arith__series,axiom,
    ! [A: nat,D3: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_if(A),D3)),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,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D3)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8176_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_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 )
       => ( aa_set_nat_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_8177_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))) = aa_rat_rat(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))) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8178_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_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 )
       => ( aa_set_nat_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_8179_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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_8180_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),aa_num_real(numeral_numeral_real,aa_num_num(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_8181_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)),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_8182_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)),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(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_8183_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8184_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,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8185_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ii,K)))) ).

% finite_Collect_le_nat
tff(fact_8186_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,K)))) ).

% finite_Collect_less_nat
tff(fact_8187_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A3))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A3)) = 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,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_8188_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),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ik(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_8189_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_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_im(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_8190_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),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_io(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_8191_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ip(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8192_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_iq(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8193_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aTP_Lamm_ir(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8194_choose__odd__sum,axiom,
    ! [N: nat] :
      ( 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_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_is(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_8195_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_it(A),B)))) ).

% finite_interval_int1
tff(fact_8196_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_iu(A),B)))) ).

% finite_interval_int4
tff(fact_8197_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or890127255671739683et_nat(K))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(I),K)) ) ).

% lessThan_iff
tff(fact_8198_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_8199_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_8200_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_8201_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_8202_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_8203_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_ord_atMost_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),K)) ) ).

% atMost_iff
tff(fact_8204_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(I),K)) ) ).

% atMost_iff
tff(fact_8205_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_atMost_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),K)) ) ).

% atMost_iff
tff(fact_8206_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_atMost_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),K)) ) ).

% atMost_iff
tff(fact_8207_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_atMost_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K)) ) ).

% atMost_iff
tff(fact_8208_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_8209_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_iv(N),C)))) ) ).

% finite_nth_roots
tff(fact_8210_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_iw(A),B)))) ).

% finite_interval_int2
tff(fact_8211_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_ix(A),B)))) ).

% finite_interval_int3
tff(fact_8212_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8213_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8214_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8215_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8216_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_8217_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X),set_or4236626031148496127et_nat(Y))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% atMost_subset_iff
tff(fact_8218_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(X)),set_ord_atMost_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8219_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(X)),set_ord_atMost_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8220_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8221_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8222_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8223_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_8224_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H),set_or4236626031148496127et_nat(H2))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
        | pp(aa_set_nat_bool(ord_less_eq_set_nat(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8225_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(H2)))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8226_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(H2)))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8227_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8228_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H2)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8229_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H2)))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8230_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_8231_int__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_ar(F),A3) ).

% int_sum
tff(fact_8232_int__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups3539618377306564664at_int(aTP_Lamm_av(F),A3) ).

% int_sum
tff(fact_8233_Complex__sum_H,axiom,
    ! [F: fun_nat_real,S: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_iy(F)),S) = aa_real_complex(complex2(aa_set_nat_real(groups6591440286371151544t_real(F),S)),zero_zero_real) ).

% Complex_sum'
tff(fact_8234_Complex__sum_H,axiom,
    ! [F: fun_complex_real,S: set_complex] : groups7754918857620584856omplex(aTP_Lamm_iz(F),S) = aa_real_complex(complex2(groups5808333547571424918x_real(F,S)),zero_zero_real) ).

% Complex_sum'
tff(fact_8235_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,N: real] :
      ( ! [X4: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X4)),aa_real_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(N))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ja(Q),P),set_or5984915006950818249n_real(N)) ) ) ).

% sum_diff_distrib
tff(fact_8236_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jb(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_8237_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(A)),set_ord_lessThan_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8238_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(A)),set_ord_lessThan_num(B)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8239_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8240_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8241_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(A)),set_or5984915006950818249n_real(B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8242_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_jc(U)) ).

% lessThan_def
tff(fact_8243_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_jd,U)) ).

% lessThan_def
tff(fact_8244_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_je,U)) ).

% lessThan_def
tff(fact_8245_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_jf,U)) ).

% lessThan_def
tff(fact_8246_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,U)) ).

% lessThan_def
tff(fact_8247_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jg,U)) ).

% lessThan_def
tff(fact_8248_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jh,U)) ).

% atMost_def
tff(fact_8249_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_ji(U)) ).

% atMost_def
tff(fact_8250_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_jj,U)) ).

% atMost_def
tff(fact_8251_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_jk,U)) ).

% atMost_def
tff(fact_8252_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_jl,U)) ).

% atMost_def
tff(fact_8253_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ii,U)) ).

% atMost_def
tff(fact_8254_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8255_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8256_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,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_bk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8257_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_8258_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8259_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8260_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8261_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_8262_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_8263_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_8264_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_8265_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_8266_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_8267_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_8268_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M7)
     => ? [N2: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X5,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_8269_finite__maxlen,axiom,
    ! [M7: set_list_bool] :
      ( finite7866827932528546246t_bool(M7)
     => ? [N2: nat] :
        ! [X5: list_bool] :
          ( member_list_bool(X5,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_8270_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( finite8100373058378681591st_nat(M7)
     => ? [N2: nat] :
        ! [X5: list_nat] :
          ( member_list_nat(X5,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_8271_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( finite3922522038869484883st_int(M7)
     => ? [N2: nat] :
        ! [X5: list_int] :
          ( member_list_int(X5,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_8272_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_8273_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_jm,collect_complex(aTP_Lamm_hb(N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8274_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_jm,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_iv(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8275_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A3: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_jn(C),F)),A3) ).

% power_sum
tff(fact_8276_power__sum,axiom,
    ! [C: complex,F: fun_nat_nat,A3: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_jo(C),F)),A3) ).

% power_sum
tff(fact_8277_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A3: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_jp(C),F),A3) ).

% power_sum
tff(fact_8278_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A3: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A3)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_jq(C),F),A3) ).

% power_sum
tff(fact_8279_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jr(C),F)),A3) ).

% power_sum
tff(fact_8280_sum__subtractf__nat,axiom,
    ! [A3: set_option_nat,G: fun_option_nat_nat,F: fun_option_nat_nat] :
      ( ! [X4: option_nat] :
          ( member_option_nat(X4,A3)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_option_nat_nat(G,X4)),aa_option_nat_nat(F,X4))) )
     => ( groups1921984841883549356at_nat(aa_fun7124324411455820056at_nat(aTP_Lamm_js(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1921984841883549356at_nat(F,A3)),groups1921984841883549356at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8281_sum__subtractf__nat,axiom,
    ! [A3: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ja(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8282_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,A3)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_jt(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A3)),groups8294997508430121362at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8283_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ju(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8284_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jb(G),F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_8285_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,N: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8286_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(C),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8287_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,N: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(C),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8288_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,N: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_8289_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,N: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8290_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,N: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(C),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8291_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,N: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(C),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8292_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,N: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_8293_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_jx,I)))) ) ).

% finite_divisors_int
tff(fact_8294_sum__eq__Suc0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups4541462559716669496nt_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8295_sum__eq__Suc0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups5693394587270226106ex_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8296_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( finite6177210948735845034at_nat(A3)
     => ( ( groups977919841031483927at_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X3,A3)
            & ( aa_Pro4243173770044397973at_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: product_prod_nat_nat] :
                ( member8440522571783428010at_nat(Xa4,A3)
               => ( ( X3 != Xa4 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8297_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( groups2027974829824023292at_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & ( aa_Extended_enat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_Extended_enat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8298_sum__eq__Suc0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8299_sum__SucD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,N) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_8300_sum__eq__1__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( groups4541462559716669496nt_nat(F,A3) = one_one_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A3))
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8301_sum__eq__1__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A3))
     => ( ( groups5693394587270226106ex_nat(F,A3) = one_one_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A3))
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8302_sum__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( finite6177210948735845034at_nat(A3)
     => ( ( groups977919841031483927at_nat(F,A3) = one_one_nat )
      <=> ? [X3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X3,A3)
            & ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat )
            & ! [Xa4: product_prod_nat_nat] :
                ( member8440522571783428010at_nat(Xa4,A3)
               => ( ( X3 != Xa4 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8303_sum__eq__1__iff,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( groups2027974829824023292at_nat(F,A3) = one_one_nat )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A3))
            & ( aa_Extended_enat_nat(F,X3) = one_one_nat )
            & ! [Xa4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_Extended_enat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8304_sum__eq__1__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = one_one_nat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A3))
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A3))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8305_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_8306_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_8307_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( aa_set_nat_nat(nat_set_encode,A3) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_8308_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_jz(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8309_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_kb(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8310_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_kd(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8311_polyfun__diff__alt,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_kf(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_8312_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [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))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8313_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( summable_nat(F)
     => ( ! [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))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8314_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X)) ) ) ).

% suminf_le_const
tff(fact_8315_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8316_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8317_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_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_bk(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8318_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_8319_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_kg(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8320_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_kh(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8321_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ki(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8322_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,D3: fun_nat_complex] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(D3),X3)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_complex(C,I4) = aa_nat_complex(D3,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8323_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,D3: fun_nat_real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(D3),X3)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_real(C,I4) = aa_nat_real(D3,I4) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8324_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B4: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N2))),B4))
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_8325_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B4: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B4))
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_8326_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B4: real] :
      ( ! [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_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N2))),B4))
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_8327_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8328_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8329_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8330_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bk(G)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_8331_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))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8332_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(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8333_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_bk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8334_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(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)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8335_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_gx(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8336_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_gy(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8337_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gz(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_8338_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_kg(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8339_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_kh(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8340_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ki(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_8341_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_complex,N: nat,R2: complex] : aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),R2)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kj(F),R2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8342_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,N: nat,R2: rat] : aa_rat_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_kk(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8343_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,N: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_kl(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8344_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,N: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_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)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_km(F),R2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8345_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_8346_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_8347_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,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8348_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8349_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_bi(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8350_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8351_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_8352_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_kn(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_8353_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_bk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_8354_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bh(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_8355_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_kp(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8356_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_kr(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8357_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_kt(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8358_polyfun__diff,axiom,
    ! [N: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_kv(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_8359_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = groups705719431365010083at_int(aTP_Lamm_bj(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_8360_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_bk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_8361_norm__prod__diff,axiom,
    ! [I5: set_option_nat,Z: fun_option_nat_real,W: fun_option_nat_real] :
      ( ! [I3: option_nat] :
          ( member_option_nat(I3,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_option_nat_real(Z,I3))),one_one_real)) )
     => ( ! [I3: option_nat] :
            ( member_option_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_option_nat_real(W,I3))),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,groups6579596140936342515t_real(Z,I5)),groups6579596140936342515t_real(W,I5)))),groups4518532050878116744t_real(aa_fun2975597672206361808t_real(aTP_Lamm_kw(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8362_norm__prod__diff,axiom,
    ! [I5: set_real,Z: fun_real_real,W: fun_real_real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(Z,I3))),one_one_real)) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(W,I3))),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,groups1681761925125756287l_real(Z,I5)),groups1681761925125756287l_real(W,I5)))),groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_kx(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8363_norm__prod__diff,axiom,
    ! [I5: set_set_nat,Z: fun_set_nat_real,W: fun_set_nat_real] :
      ( ! [I3: set_nat] :
          ( member_set_nat(I3,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(Z,I3))),one_one_real)) )
     => ( ! [I3: set_nat] :
            ( member_set_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(W,I3))),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,groups3619160379726066777t_real(Z,I5)),groups3619160379726066777t_real(W,I5)))),groups5107569545109728110t_real(aa_fun3569254276315353500t_real(aTP_Lamm_ky(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8364_norm__prod__diff,axiom,
    ! [I5: set_int,Z: fun_int_real,W: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(Z,I3))),one_one_real)) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(W,I3))),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,groups2316167850115554303t_real(Z,I5)),groups2316167850115554303t_real(W,I5)))),groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_kz(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8365_norm__prod__diff,axiom,
    ! [I5: set_option_nat,Z: fun_op5246294523634652747omplex,W: fun_op5246294523634652747omplex] :
      ( ! [I3: option_nat] :
          ( member_option_nat(I3,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_opt3066276567374697100omplex(Z,I3))),one_one_real)) )
     => ( ! [I3: option_nat] :
            ( member_option_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_opt3066276567374697100omplex(W,I3))),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(groups1917955009447795573omplex(Z,I5)),groups1917955009447795573omplex(W,I5)))),groups4518532050878116744t_real(aa_fun240986247698212690t_real(aTP_Lamm_la(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8366_norm__prod__diff,axiom,
    ! [I5: set_real,Z: fun_real_complex,W: fun_real_complex] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(Z,I3))),one_one_real)) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(W,I3))),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(groups713298508707869441omplex(Z,I5)),groups713298508707869441omplex(W,I5)))),groups8097168146408367636l_real(aa_fun2786211375485122666l_real(aTP_Lamm_lb(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8367_norm__prod__diff,axiom,
    ! [I5: set_set_nat,Z: fun_set_nat_complex,W: fun_set_nat_complex] :
      ( ! [I3: set_nat] :
          ( member_set_nat(I3,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(Z,I3))),one_one_real)) )
     => ( ! [I3: set_nat] :
            ( member_set_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(W,I3))),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(groups1092910753850256091omplex(Z,I5)),groups1092910753850256091omplex(W,I5)))),groups5107569545109728110t_real(aa_fun5767801665285160478t_real(aTP_Lamm_lc(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8368_norm__prod__diff,axiom,
    ! [I5: set_int,Z: fun_int_complex,W: fun_int_complex] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(Z,I3))),one_one_real)) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(W,I3))),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(groups7440179247065528705omplex(Z,I5)),groups7440179247065528705omplex(W,I5)))),groups8778361861064173332t_real(aa_fun1985759105522019690t_real(aTP_Lamm_ld(Z),W),I5))) ) ) ).

% norm_prod_diff
tff(fact_8369_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: fun_nat_real,W: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Z,I3))),one_one_real)) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(W,I3))),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_set_nat_real(groups129246275422532515t_real(Z),I5)),aa_set_nat_real(groups129246275422532515t_real(W),I5)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_le(Z),W)),I5))) ) ) ).

% norm_prod_diff
tff(fact_8370_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: fun_nat_complex,W: fun_nat_complex] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Z,I3))),one_one_real)) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(W,I3))),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_set_nat_complex(groups6464643781859351333omplex(Z),I5)),aa_set_nat_complex(groups6464643781859351333omplex(W),I5)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun7054779078796324018t_real(aTP_Lamm_lf(Z),W)),I5))) ) ) ).

% norm_prod_diff
tff(fact_8371_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,N: nat,K: nat] :
      ( ! [W2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8372_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,N: nat,K: nat] :
      ( ! [W2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8373_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_complex(C,I4) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8374_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
         => ( aa_nat_real(C,I4) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8375_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)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8376_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(aa_rat_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_8377_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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_8378_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)),aa_set_nat_real(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_8379_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)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8380_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_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_8381_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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_8382_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)),aa_set_nat_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_8383_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( aa_set_nat_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_8384_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)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N)),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8385_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_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_8386_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,aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_8387_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_lg(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_8388_ln__prod,axiom,
    ! [I5: set_option_nat,F: fun_option_nat_real] :
      ( finite5523153139673422903on_nat(I5)
     => ( ! [I3: option_nat] :
            ( member_option_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_option_nat_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups6579596140936342515t_real(F,I5)) = groups4518532050878116744t_real(aTP_Lamm_lh(F),I5) ) ) ) ).

% ln_prod
tff(fact_8389_ln__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ! [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))) )
       => ( aa_real_real(ln_ln_real,groups1681761925125756287l_real(F,I5)) = groups8097168146408367636l_real(aTP_Lamm_li(F),I5) ) ) ) ).

% ln_prod
tff(fact_8390_ln__prod,axiom,
    ! [I5: set_set_nat,F: fun_set_nat_real] :
      ( finite1152437895449049373et_nat(I5)
     => ( ! [I3: set_nat] :
            ( member_set_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_set_nat_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups3619160379726066777t_real(F,I5)) = groups5107569545109728110t_real(aTP_Lamm_lj(F),I5) ) ) ) ).

% ln_prod
tff(fact_8391_ln__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ! [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))) )
       => ( aa_real_real(ln_ln_real,groups2316167850115554303t_real(F,I5)) = groups8778361861064173332t_real(aTP_Lamm_lk(F),I5) ) ) ) ).

% ln_prod
tff(fact_8392_ln__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups766887009212190081x_real(F,I5)) = groups5808333547571424918x_real(aTP_Lamm_ll(F),I5) ) ) ) ).

% ln_prod
tff(fact_8393_ln__prod,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( finite6177210948735845034at_nat(I5)
     => ( ! [I3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(I3,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Pro6168199777849092593t_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups6036352826371341000t_real(F,I5)) = groups4567486121110086003t_real(aTP_Lamm_lm(F),I5) ) ) ) ).

% ln_prod
tff(fact_8394_ln__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ! [I3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,groups97031904164794029t_real(F,I5)) = groups4148127829035722712t_real(aTP_Lamm_ln(F),I5) ) ) ) ).

% ln_prod
tff(fact_8395_ln__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I3))) )
       => ( aa_real_real(ln_ln_real,aa_set_nat_real(groups129246275422532515t_real(F),I5)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lo(F)),I5) ) ) ) ).

% ln_prod
tff(fact_8396_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M4))) )
       => 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_8397_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M4))) )
       => 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_8398_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M4))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_8399_sum__gp__basic,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8400_sum__gp__basic,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8401_sum__gp__basic,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_8402_sum__gp__basic,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = 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))) ).

% sum_gp_basic
tff(fact_8403_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,N: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_lp(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_8404_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,N: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_lq(C),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_8405_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_lp(C),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_complex(C,I4) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8406_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_lq(C),N))))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),N))
          & ( aa_nat_real(C,I4) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8407_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_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_8408_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)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_strict
tff(fact_8409_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_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 )
       => ( aa_set_nat_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_8410_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lr(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ls(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8411_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lt(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lu(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8412_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lv(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lw(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8413_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lx(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ly(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8414_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)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_lz(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8415_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_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(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ma(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8416_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_mb(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8417_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)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mc(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8418_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)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_md(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8419_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_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(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_me(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8420_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_mf(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8421_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)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mg(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8422_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 ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_nat(B,J2) = 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_mh(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(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_mj(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_8423_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8424_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8425_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8426_sum__power__shift,axiom,
    ! [M: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_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(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% sum_power_shift
tff(fact_8427_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk,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,aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_8428_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_8429_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_8430_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K5: 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)),K5)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K5))
       => 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)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8431_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K5: 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)),K5)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K5))
       => 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(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8432_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K5: 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)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K5))
       => 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)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8433_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K5: 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)),K5)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K5))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_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)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8434_finite__has__minimal2,axiom,
    ! [A3: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(A),A3))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8435_finite__has__minimal2,axiom,
    ! [A3: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
            & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),A))
            & ! [Xa2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8436_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( member_set_nat(A,A3)
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A3)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),A))
            & ! [Xa2: set_nat] :
                ( member_set_nat(Xa2,A3)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8437_finite__has__minimal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(member_rat(A),A3))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A3))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),A))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8438_finite__has__minimal2,axiom,
    ! [A3: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(member_num(A),A3))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A3))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),A))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8439_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(A),A3))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),A))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8440_finite__has__minimal2,axiom,
    ! [A3: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(A),A3))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),A))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8441_finite__has__maximal2,axiom,
    ! [A3: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( pp(aa_set_real_bool(member_real(A),A3))
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8442_finite__has__maximal2,axiom,
    ! [A3: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A3))
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
            & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),X4))
            & ! [Xa2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8443_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( member_set_nat(A,A3)
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A3)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X4))
            & ! [Xa2: set_nat] :
                ( member_set_nat(Xa2,A3)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8444_finite__has__maximal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( pp(aa_set_rat_bool(member_rat(A),A3))
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A3))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X4))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8445_finite__has__maximal2,axiom,
    ! [A3: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( pp(aa_set_num_bool(member_num(A),A3))
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A3))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X4))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8446_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_set_nat_bool(member_nat(A),A3))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8447_finite__has__maximal2,axiom,
    ! [A3: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( pp(aa_set_int_bool(member_int(A),A3))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8448_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M4))) )
       => ( 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_8449_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M4))) )
       => ( 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_8450_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M4))) )
       => ( 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,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8451_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_hx(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8452_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_hy(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8453_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hz(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8454_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ia(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_8455_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_complex,N: nat,B: fun_nat_complex,X: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_complex(A,I3) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_mm(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
tff(fact_8456_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_rat,N: nat,B: fun_nat_rat,X: rat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_rat(A,I3) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(B),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_mo(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
tff(fact_8457_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_int,N: nat,B: fun_nat_int,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_int(A,I3) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jw(B),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_mq(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
tff(fact_8458_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_real,N: nat,B: fun_nat_real,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_real(A,I3) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_ms(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
tff(fact_8459_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_bv(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8460_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_bw(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8461_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_bx(G),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8462_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_by(G)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_8463_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,N: nat,K: complex] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8464_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,N: nat,K: real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8465_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)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_mt(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8466_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_mu(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8467_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_mv(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8468_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)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mw(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8469_binomial__ring,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),N) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mx(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8470_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_my(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8471_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_mz(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8472_binomial__ring,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_lg(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8473_binomial__ring,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_na(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_8474_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B)),N) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nb(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8475_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),B)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nc(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8476_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nd(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8477_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ne(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_8478_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nf(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ng(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8479_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nh(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ni(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8480_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nj(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nk(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8481_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_nl(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_nm(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8482_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_nn(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_no(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8483_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_np(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_nm(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8484_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nq(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ng(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8485_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nr(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ni(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8486_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ns(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nk(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8487_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,aa_num_num(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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_8488_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nt,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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_8489_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nu(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,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nv(F)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nw(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_8490_root__polyfun,axiom,
    ! [N: nat,Z: int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_int(power_power_int(Z),N) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_nx(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8491_root__polyfun,axiom,
    ! [N: nat,Z: complex,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_complex(power_power_complex(Z),N) = A )
      <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ny(N),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8492_root__polyfun,axiom,
    ! [N: nat,Z: code_integer,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),N) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_nz(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8493_root__polyfun,axiom,
    ! [N: nat,Z: rat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_rat(power_power_rat(Z),N) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_oa(N),Z),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8494_root__polyfun,axiom,
    ! [N: nat,Z: real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ob(N),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8495_sum__gp0,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(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_gp0
tff(fact_8496_sum__gp0,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp0
tff(fact_8497_sum__gp0,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_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),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp0
tff(fact_8498_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_oc(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8499_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_od(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8500_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_oe(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8501_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups3539618377306564664at_int(aTP_Lamm_of(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8502_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_og(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8503_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups7501900531339628137nteger(aTP_Lamm_oh(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8504_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_oi(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8505_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_oj(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8506_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_ok(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8507_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ol(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8508_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8509_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8510_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A3)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A3)) ) ) ).

% even_set_encode_iff
tff(fact_8511_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D6: 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))),D6)))),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))),D6)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_8512_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_cc,set_or1266510415728281911st_int(M,N)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_8513_finite__has__minimal,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
            & ! [Xa2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8514_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8515_finite__has__minimal,axiom,
    ! [A3: set_set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A3)
            & ! [Xa2: set_nat] :
                ( member_set_nat(Xa2,A3)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8516_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( ( A3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A3))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8517_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( ( A3 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A3))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8518_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( A3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A3))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8519_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X4))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8520_finite__has__maximal,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A3))
            & ! [Xa2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa2),A3))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8521_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A3))
     => ( ( A3 != bot_bot_set_real )
       => ? [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8522_finite__has__maximal,axiom,
    ! [A3: set_set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A3)
            & ! [Xa2: set_nat] :
                ( member_set_nat(Xa2,A3)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8523_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A3))
     => ( ( A3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),A3))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A3))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8524_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A3))
     => ( ( A3 != bot_bot_set_num )
       => ? [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),A3))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A3))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8525_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A3))
     => ( ( A3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A3))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A3))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8526_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A3))
     => ( ( A3 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A3))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A3))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa2))
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8527_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_om(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8528_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_on(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8529_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aTP_Lamm_oo(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8530_choose__even__sum,axiom,
    ! [N: nat] :
      ( 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_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_op(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_8531_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oq),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8532_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_or(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8533_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_os(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8534_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ot,A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8535_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,M))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8536_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8537_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8538_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8539_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8540_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8541_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8542_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,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_8543_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_8544_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_8545_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_8546_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_8547_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8548_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8549_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8550_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_8551_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8552_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_8553_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8554_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_8555_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_8556_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_8557_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_8558_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_8559_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_8560_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_8561_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_8562_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_8563_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8564_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8565_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8566_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_8567_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_8568_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_8569_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_8570_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).

% dbl_dec_simps(1)
tff(fact_8571_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_8572_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_8573_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_8574_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_8575_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).

% dbl_inc_simps(1)
tff(fact_8576_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_8577_cos__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))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_8578_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_8579_of__nat__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,N),K)) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),K) ).

% of_nat_gbinomial
tff(fact_8580_of__nat__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,N),K)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K) ).

% of_nat_gbinomial
tff(fact_8581_of__nat__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,N),K)) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) ).

% of_nat_gbinomial
tff(fact_8582_binomial__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),K) ).

% binomial_gbinomial
tff(fact_8583_binomial__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K) ).

% binomial_gbinomial
tff(fact_8584_binomial__gbinomial,axiom,
    ! [N: nat,K: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) ).

% binomial_gbinomial
tff(fact_8585_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),K) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8586_gbinomial__of__nat__symmetric,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(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8587_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8588_arccos__le__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_eq_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_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_8589_arccos__eq__iff,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)) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_8590_arccos__le__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_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% arccos_le_mono
tff(fact_8591_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_8592_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_8593_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_8594_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8595_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8596_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8597_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8598_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8599_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8600_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
     => 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(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8601_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8602_dbl__inc__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_inc_def
tff(fact_8603_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_8604_dbl__inc__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_8605_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_8606_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_8607_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_8608_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_8609_arccos__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(arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_8610_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_8611_cos__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(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_8612_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi))
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_8613_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_8614_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% Suc_times_gbinomial
tff(fact_8615_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_8616_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_8617_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_8618_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8619_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),M)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8620_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),M)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8621_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8622_gbinomial__parallel__sum,axiom,
    ! [A: complex,N: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ou(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),N) ).

% gbinomial_parallel_sum
tff(fact_8623_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_ov(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),N) ).

% gbinomial_parallel_sum
tff(fact_8624_gbinomial__parallel__sum,axiom,
    ! [A: real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ow,A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),N) ).

% gbinomial_parallel_sum
tff(fact_8625_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_8626_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_8627_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_8628_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% gbinomial_factors
tff(fact_8629_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_8630_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8631_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8632_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8633_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),N)) ).

% gbinomial_index_swap
tff(fact_8634_gbinomial__index__swap,axiom,
    ! [K: nat,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)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),K)) = 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(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),N)) ).

% gbinomial_index_swap
tff(fact_8635_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),N)) ).

% gbinomial_index_swap
tff(fact_8636_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_8637_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_8638_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_8639_arccos__minus,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(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_8640_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_8641_arccos__minus__abs,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(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_8642_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_8643_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_8644_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,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_8645_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))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ox,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8646_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))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ox,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8647_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ox,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8648_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M3),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8649_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,aa_num_num(bit0,one))),N)))) ).

% square_fact_le_2_fact
tff(fact_8650_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8651_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M4))
         => ? [N8: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N8))
              & pp(aa_set_nat_bool(member_nat(N8),S2)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).

% unbounded_k_infinite
tff(fact_8652_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N3))
          & pp(aa_set_nat_bool(member_nat(N3),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_8653_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B6: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_oy(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B6),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_8654_cos__coeff__def,axiom,
    ! [X5: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(cos_coeff,X5) = 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,X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(cos_coeff,X5) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_8655_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N3)))
          & pp(aa_set_int_bool(member_int(N3),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_8656_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))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oz,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8657_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oz,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8658_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oz,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8659_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] : aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oz,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,T2),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),aa_num_real(numeral_numeral_real,aa_num_num(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_8660_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8661_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_8662_fact__mono__nat,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(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_8663_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_8664_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_8665_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_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(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_8666_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_8667_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_8668_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_8669_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)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ib),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8670_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_8671_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)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ib),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_8672_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_8673_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_8674_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,aa_num_num(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,aa_num_num(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_8675_sin__coeff__def,axiom,
    ! [X5: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(sin_coeff,X5) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(sin_coeff,X5) = 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,X5),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5)) ) ) ) ).

% sin_coeff_def
tff(fact_8676_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))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pa,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,T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8677_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_pb,X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_8678_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_8679_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_8680_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_8681_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% exp_le_cancel_iff
tff(fact_8682_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_8683_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_8684_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_8685_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_8686_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_8687_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_8688_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_8689_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_8690_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_8691_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_8692_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] : aa_real_real(exp_real,X4) = Y ) ).

% exp_total
tff(fact_8693_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_8694_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_8695_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_8696_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_8697_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_8698_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real)))
          & ( aa_real_real(exp_real,X4) = Y ) ) ) ).

% lemma_exp_total
tff(fact_8699_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_8700_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_8701_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_8702_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% exp_half_le2
tff(fact_8703_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_pc),one_one_real)) ).

% power_half_series
tff(fact_8704_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_pd(G)),X)) ) ).

% sums_if'
tff(fact_8705_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_pe(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sums_if
tff(fact_8706_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,aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_8707_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),aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_8708_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_8709_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_8710_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pa,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,T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_8711_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,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_8712_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_8713_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,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_8714_cos__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_pf,X)),aa_real_real(cos_real,X))) ).

% cos_paired
tff(fact_8715_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_8716_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)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oz,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_8717_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_8718_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_8719_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_8720_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_8721_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_8722_semiring__norm_I28_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).

% semiring_norm(28)
tff(fact_8723_semiring__norm_I27_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(bitM,N)) ).

% semiring_norm(27)
tff(fact_8724_inc__BitM__eq,axiom,
    ! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_8725_BitM__inc__eq,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_8726_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_8727_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_8728_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_8729_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_8730_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D6: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D6),E))
         => ( pp(aa_real_bool(P,D6))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [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),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_8731_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D6: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D6),E))
         => ( pp(aa_real_bool(P,D6))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [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),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_8732_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N3: nat] :
          ( ( N3 != 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,N3))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))),E2)) ) ) ).

% real_arch_inverse
tff(fact_8733_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_8734_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_8735_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_8736_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(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_8737_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,aa_num_real(numeral_numeral_real,aa_num_num(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_8738_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,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8739_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_8740_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_8741_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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_8742_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_8743_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,aa_num_num(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),aa_num_num(bitM,N)))) ).

% or_minus_numerals(5)
tff(fact_8744_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,aa_num_num(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),aa_num_num(bitM,N)))) ).

% or_minus_numerals(1)
tff(fact_8745_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_8746_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_8747_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% sinh_real_le_iff
tff(fact_8748_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_8749_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_8750_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_8751_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_8752_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),aa_num_num(bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_8753_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),aa_num_num(bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_8754_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,aa_num_num(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),aa_num_num(bitM,N)))) ).

% or_minus_numerals(3)
tff(fact_8755_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,aa_num_num(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),aa_num_num(bitM,N)))) ).

% or_minus_numerals(7)
tff(fact_8756_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_8757_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_le_cosh_real
tff(fact_8758_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_8759_cosh__real__nonzero,axiom,
    ! [X: real] : aa_real_real(cosh_real,X) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_8760_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_8761_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_8762_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_8763_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_8764_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X))) ).

% cosh_real_ge_1
tff(fact_8765_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_8766_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,aa_num_num(bit0,N)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(6)
tff(fact_8767_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_8768_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_8769_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,aa_num_num(bit0,N)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(5)
tff(fact_8770_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)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(9)
tff(fact_8771_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_8772_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_8773_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_8774_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_8775_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_8776_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)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(8)
tff(fact_8777_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X),Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M4: num] :
                ( ( Xa = aa_num_num(bit0,M4) )
               => ( Y != aa_num_num(bit1,M4) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa = aa_num_num(bit1,M4) )
                 => ( Y != aa_num_num(bit1,M4) ) ) )
           => ( ( ? [N2: num] : X = aa_num_num(bit0,N2)
               => ( ( Xa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa_num_num(bit0,N2) )
                   => ! [M4: num] :
                        ( ( Xa = aa_num_num(bit0,M4) )
                       => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa_num_num(bit0,N2) )
                     => ! [M4: num] :
                          ( ( Xa = aa_num_num(bit1,M4) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) ) ) )
                 => ( ( ? [N2: num] : X = aa_num_num(bit1,N2)
                     => ( ( Xa = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa_num_num(bit1,N2) )
                         => ! [M4: num] :
                              ( ( Xa = aa_num_num(bit0,M4) )
                             => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X = aa_num_num(bit1,N2) )
                           => ! [M4: num] :
                                ( ( Xa = aa_num_num(bit1,M4) )
                               => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_8778_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_8779_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_8780_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_8781_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_8782_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8783_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8784_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(N,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_8785_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)),aa_num_real(numeral_numeral_real,aa_num_num(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_8786_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,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_8787_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_8788_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_8789_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_8790_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_8791_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_8792_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_8793_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_8794_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_8795_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_8796_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(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_8797_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_8798_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_8799_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_8800_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_8801_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_8802_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_8803_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_8804_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_8805_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_pg(L)),Qr) ).

% divmod_step_nat_def
tff(fact_8806_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ph(L),Qr) ).

% divmod_step_int_def
tff(fact_8807_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),aa_num_real(numeral_numeral_real,aa_num_num(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_8808_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_8809_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_8810_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_8811_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_8812_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_8813_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
tff(fact_8814_csqrt__0,axiom,
    aa_complex_complex(csqrt,zero_zero_complex) = zero_zero_complex ).

% csqrt_0
tff(fact_8815_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_8816_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_8817_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_8818_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_8819_cis__neq__zero,axiom,
    ! [A: real] : aa_real_complex(cis,A) != zero_zero_complex ).

% cis_neq_zero
tff(fact_8820_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_8821_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_8822_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_8823_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_8824_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) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_pi),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_8825_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_pj(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aTP_Lamm_hb(N)))) ) ).

% bij_betw_roots_unity
tff(fact_8826_nat__of__bool,axiom,
    ! [P: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n2687167440665602831ol_nat(P) ).

% nat_of_bool
tff(fact_8827_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) = 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_8828_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)) = 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_8829_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] : adjust_div(product_Pair_int_int(Q3,R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q3),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_8830_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_pk,Qr) ).

% Divides.adjust_div_def
tff(fact_8831_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_8832_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),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_8833_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),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_8834_or__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_8835_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,zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_8836_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,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,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_8837_xor__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_8838_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),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_8839_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),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_8840_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,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(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(aa_int_fun_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,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_8841_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa) = 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(Xa),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,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))) ) )
        & ( ~ ( 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(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_8842_int__ge__less__than2__def,axiom,
    ! [D3: int] : int_ge_less_than2(D3) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_pl(D3))) ).

% int_ge_less_than2_def
tff(fact_8843_int__ge__less__than__def,axiom,
    ! [D3: int] : int_ge_less_than(D3) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_pm(D3))) ).

% int_ge_less_than_def
tff(fact_8844_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_8845_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_8846_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_8847_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_8848_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,aa_num_num(bit0,N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_8849_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,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_8850_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(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_8851_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(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_8852_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_8853_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_8854_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_8855_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_8856_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_8857_plus__and__or,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),aa_int_int(bit_se1409905431419307370or_int(X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) ).

% plus_and_or
tff(fact_8858_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_8859_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_8860_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_8861_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_8862_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_8863_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_8864_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,aa_num_num(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,aa_num_num(bit0,one))),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)) ) ) ).

% even_and_iff_int
tff(fact_8865_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,aa_num_num(bit0,N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_8866_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_8867_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_8868_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,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_8869_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_8870_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_8871_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_8872_and__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_8873_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,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(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_8874_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_8875_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))
       => pp(aa_set_complex_bool(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_hb(N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_pn(C),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8876_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( 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,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(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(aa_int_fun_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,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_8877_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa))
       => ~ ( ( ( ( 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(Xa),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,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))) ) )
              & ( ~ ( 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(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(aa_int_fun_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,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa)) ) ) ) ).

% and_int.pelims
tff(fact_8878_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,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = 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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = 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,aa_num_num(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,aa_num_num(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,aa_num_num(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,aa_num_num(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_8879_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_8880_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_8881_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_8882_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_8883_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_8884_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_8885_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_8886_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_8887_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_8888_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_8889_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_8890_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_8891_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_8892_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_8893_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_8894_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_8895_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_8896_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,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_8897_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,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_8898_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_8899_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_8900_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_8901_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,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_8902_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,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_8903_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_8904_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_8905_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_8906_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_8907_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_8908_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_8909_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_8910_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_8911_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_8912_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_8913_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_8914_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_8915_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_8916_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_8917_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_8918_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_8919_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_8920_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_8921_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_8922_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_8923_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,aa_num_num(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_8924_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,aa_num_num(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_8925_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,aa_num_num(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_8926_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_8927_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_8928_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_8929_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_8930_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_8931_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_8932_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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_8933_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_8934_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,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,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_8935_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L3: int] :
            ( 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(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(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_8936_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I3,J2))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_8937_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,aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_8938_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,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_8939_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_8940_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_8941_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,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_8942_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_po,X)) ).

% set_decode_def
tff(fact_8943_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pp,Y)) ).

% arctan_def
tff(fact_8944_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pq,Y)) ).

% arcsin_def
tff(fact_8945_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_pr) ) ) ).

% ln_neg_is_const
tff(fact_8946_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ps,Y)) ).

% arccos_def
tff(fact_8947_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_pt) ).

% pi_half
tff(fact_8948_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_pt)) ).

% pi_def
tff(fact_8949_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(aa_int_fun_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(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_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_8950_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_8951_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_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(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(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)))),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_8952_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(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_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(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_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(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),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_8953_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(aa_int_fun_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_8954_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(aa_int_fun_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_8955_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(aa_int_fun_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_8956_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(aa_int_fun_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_8957_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(bit_se1146084159140164899it_int(K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_8958_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(bit_se1146084159140164899it_int(K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_8959_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,N)))
    <=> pp(aa_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_8960_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_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(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_8961_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_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_8962_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_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(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_8963_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(K),L)),N))
    <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).

% bit_and_int_iff
tff(fact_8964_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(K),L)),N))
    <=> ~ ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
        <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_8965_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),N))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% bit_not_int_iff
tff(fact_8966_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se1409905431419307370or_int(K),L)),N))
    <=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
        | pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).

% bit_or_int_iff
tff(fact_8967_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_8968_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).

% bit_not_int_iff'
tff(fact_8969_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_8970_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_8971_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_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_8972_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(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_8973_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_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(bit_se1146084159140164899it_int(K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_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_8974_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),N))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N)) ) ).

% bit_minus_int_iff
tff(fact_8975_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(bit_se1146084159140164899it_int(K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_8976_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(bit_se1146084159140164899it_int(K),M2))
              <=> pp(aa_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(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_8977_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_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,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).

% bit_int_def
tff(fact_8978_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: 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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q3,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8979_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( 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,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8980_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q4: int] :
              ( ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22) ) ) )
         => ~ ! [R: int,Q4: int] :
                ( ( A32 = product_Pair_int_int(Q4,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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8981_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(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_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_8982_set__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% set_bit_eq
tff(fact_8983_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% unset_bit_eq
tff(fact_8984_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_Suc_from_most
tff(fact_8985_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_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(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_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)),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_8986_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_8987_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_8988_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)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_8989_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_8990_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_8991_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_8992_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_8993_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_8994_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_8995_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_8996_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_8997_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_8998_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_8999_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_9000_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_9001_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_complex(cis,aa_complex_real(arg,Z)) = aa_complex_complex(sgn_sgn_complex,Z) ) ) ).

% cis_Arg
tff(fact_9002_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_9003_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(bit_se1146084159140164899it_int(K),N)) ) ) ).

% bit_nat_iff
tff(fact_9004_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_9005_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(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9006_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9007_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,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_9008_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,aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_9009_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,aa_num_num(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,aa_num_num(bit0,one))),N)))) ) ).

% bit_nat_def
tff(fact_9010_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_9011_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_9012_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_9013_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))
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),N)) = X )
             => pp(aa_real_bool(P,Y5)) ) ) ) ) ).

% split_root
tff(fact_9014_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,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_9015_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(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_9016_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% mask_int_def
tff(fact_9017_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_9018_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,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9019_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_pu(Z)) ) ) ) ).

% Arg_def
tff(fact_9020_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9021_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,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_9022_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_9023_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_9024_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_9025_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,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_9026_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_9027_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_9028_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_9029_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_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(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9030_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: 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),Q3)),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,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9031_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(aa_int_fun_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_9032_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_9033_set__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)) ).

% set_bit_int_def
tff(fact_9034_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_9035_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_9036_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_9037_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,aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_9038_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(N),K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_int_def
tff(fact_9039_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,aa_num_num(bit0,one))),N)) ).

% push_bit_minus_one
tff(fact_9040_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9041_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9042_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).

% int_of_nat_def
tff(fact_9043_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M3))
         => ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N3))
             => 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,M3)),aa_nat_real(X7,N3)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9044_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ib),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,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9045_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
          & pp(aa_nat_bool(P,M3)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_9046_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
         => pp(aa_nat_bool(P,M3)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_9047_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_9048_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9049_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_9050_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_9051_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_9052_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_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_9053_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_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_9054_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_9055_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_9056_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,aa_num_num(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,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9057_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2))) ) )
     => ( ! [I3: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J2)),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_pv(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_9058_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_9059_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,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_9060_valid__eq,axiom,
    ! [T: vEBT_VEBT,D3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D3))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D3)) ) ).

% valid_eq
tff(fact_9061_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D3: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D3))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D3)) ) ).

% valid_eq1
tff(fact_9062_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D3))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D3)) ) ).

% valid_eq2
tff(fact_9063_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,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_9064_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9065_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_9066_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9067_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_pw(L),Qr) ).

% divmod_step_integer_def
tff(fact_9068_complex__Re__of__nat,axiom,
    ! [N: nat] : aa_complex_real(re,aa_nat_complex(semiri8010041392384452111omplex,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% complex_Re_of_nat
tff(fact_9069_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = aa_num_real(numeral_numeral_real,V) ).

% complex_Re_numeral
tff(fact_9070_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_9071_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)),aa_num_real(numeral_numeral_real,W)) ).

% Re_divide_numeral
tff(fact_9072_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_9073_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_9074_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_9075_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9076_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9077_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9078_less__eq__integer__code_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(1)
tff(fact_9079_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9080_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9081_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_9082_complex__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% complex_Re_le_cmod
tff(fact_9083_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_9084_abs__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% abs_Re_le_cmod
tff(fact_9085_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_9086_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9087_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9088_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_9089_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_9090_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))),aa_num_real(numeral_numeral_real,aa_num_num(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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9091_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9092_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) = aa_Cod1783974652533396728nteger(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,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9093_complex__Im__fact,axiom,
    ! [N: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,N)) = zero_zero_real ).

% complex_Im_fact
tff(fact_9094_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_9095_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_9096_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_9097_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9098_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_9099_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_9100_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)),aa_num_real(numeral_numeral_real,W)) ).

% Im_divide_numeral
tff(fact_9101_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_9102_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_9103_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_9104_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_9105_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9106_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_9107_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = K ) ) ) ).

% abs_integer_code
tff(fact_9108_less__integer__code_I1_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_integer_code(1)
tff(fact_9109_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X)) ) ).

% less_integer.abs_eq
tff(fact_9110_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_9111_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9112_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_9113_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(Z),ring_1_Ints_complex))
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I4: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I4) ) ) ).

% complex_is_Int_iff
tff(fact_9114_abs__Im__le__cmod,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% abs_Im_le_cmod
tff(fact_9115_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_9116_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_9117_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_9118_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_real(im,X) = aa_complex_real(im,Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_real_real(abs_abs_real,aa_complex_real(re,Y)))) ) ) ).

% cmod_Re_le_iff
tff(fact_9119_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_real(re,X) = aa_complex_real(re,Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_real_real(abs_abs_real,aa_complex_real(im,Y)))) ) ) ).

% cmod_Im_le_iff
tff(fact_9120_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_9121_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_9122_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_9123_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,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9124_Im__power2,axiom,
    ! [X: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).

% Im_power2
tff(fact_9125_Re__power2,axiom,
    ! [X: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9126_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9127_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9128_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9129_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% complex_neq_0
tff(fact_9130_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9131_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(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_9132_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,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9133_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9134_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9135_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ).

% complex_abs_le_norm
tff(fact_9136_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,aa_num_num(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,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9137_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9138_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9139_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( pp(aa_set_complex_bool(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,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9140_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( pp(aa_set_complex_bool(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,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9141_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,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9142_complex__cnj__zero,axiom,
    aa_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

% complex_cnj_zero
tff(fact_9143_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
tff(fact_9144_complex__cnj__power,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(cnj,aa_nat_complex(power_power_complex(X),N)) = aa_nat_complex(power_power_complex(aa_complex_complex(cnj,X)),N) ).

% complex_cnj_power
tff(fact_9145_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_9146_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa_set_complex_bool(member_complex(Y),real_V2521375963428798218omplex))
     => ( pp(aa_set_complex_bool(member_complex(X),real_V2521375963428798218omplex))
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = X )
        <=> ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9147_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa_set_complex_bool(member_complex(Y),real_V2521375963428798218omplex))
     => ( pp(aa_set_complex_bool(member_complex(X),real_V2521375963428798218omplex))
       => ( ( X = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9148_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(Z),real_V2521375963428798218omplex))
    <=> ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_9149_Complex__in__Reals,axiom,
    ! [X: real] : pp(aa_set_complex_bool(member_complex(aa_real_complex(complex2(X),zero_zero_real)),real_V2521375963428798218omplex)) ).

% Complex_in_Reals
tff(fact_9150_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_9151_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_9152_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_9153_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_9154_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_9155_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_9156_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_9157_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_9158_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_9159_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_9160_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,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9161_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_9162_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,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9163_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9164_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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9165_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,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9166_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,aa_num_real(numeral_numeral_real,aa_num_num(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_9167_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_int(code_int_of_integer,K) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( aa_Code_integer_int(code_int_of_integer,K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( aa_Code_integer_int(code_int_of_integer,K) = produc1553301316500091796er_int(aTP_Lamm_px,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9168_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( aa_Code_integer_num(code_num_of_integer,K) = one ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( aa_Code_integer_num(code_num_of_integer,K) = produc7336495610019696514er_num(aTP_Lamm_py,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_9169_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(S),M))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Listy)))
           => pp(aa_nat_bool(vEBT_invar_vebt(X4),N)) )
       => ( ( M = aa_nat_nat(suc,N) )
         => ( ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Listy)))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X4)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9170_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(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)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList)))))) ) ).

% height_compose_list
tff(fact_9171_int__of__integer__of__nat,axiom,
    ! [N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% int_of_integer_of_nat
tff(fact_9172_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),aa_set_nat_nat(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_9173_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),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))) ) ).

% height_i_max
tff(fact_9174_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)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))))) ) ).

% max_idx_list
tff(fact_9175_zero__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9176_int__of__integer__numeral,axiom,
    ! [K: num] : aa_Code_integer_int(code_int_of_integer,numera6620942414471956472nteger(K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9177_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gw,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9178_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,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L))) ) ).

% integer_less_iff
tff(fact_9179_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa))) ) ).

% less_integer.rep_eq
tff(fact_9180_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L))) ) ).

% integer_less_eq_iff
tff(fact_9181_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa))) ) ).

% less_eq_integer.rep_eq
tff(fact_9182_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Uu,Deg2,TreeList2,Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9183_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) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_pz(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9184_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K) = produc1555791787009142072er_nat(aTP_Lamm_qa,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_9185_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,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = zero_zero_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf(A4,B3))) ) )
         => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_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),vEBT_Node(Uu,Deg2,TreeList2,Summary2))) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9186_bij__betw__Suc,axiom,
    ! [M7: set_nat,N4: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M7),N4))
    <=> ( image_nat_nat(suc,M7) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9187_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_jx,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9188_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_Code_integer_nat(code_nat_of_integer,K)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K) ).

% of_nat_of_integer
tff(fact_9189_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9190_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A3))) ).

% zero_notin_Suc_image
tff(fact_9191_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_9192_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_9193_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9194_nat__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_nat(code_nat_of_integer,X) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% nat_of_integer.rep_eq
tff(fact_9195_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Code_integer_nat(code_nat_of_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(nat2,X) ).

% nat_of_integer.abs_eq
tff(fact_9196_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : aa_Code_integer_nat(code_nat_of_integer,numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_9197_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_9198_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_9199_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_9200_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_9201_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_qb(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9202_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_9203_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_qc,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_qc,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_qc,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9204_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,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( ( pp(B3)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( ( 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,B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9205_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,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( ( pp(A4)
                   => ( Y = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A4)
                   => ( ( pp(B3)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A4,B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9206_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: 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,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( 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,B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = 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),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9207_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,N))) = N ).

% card_Collect_less_nat
tff(fact_9208_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ii,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_9209_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_9210_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9211_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_qd),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_9212_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_qe),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_9213_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_qf(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_qg(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_9214_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_qf(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_qg(M7),I))) ) ) ).

% card_less_Suc
tff(fact_9215_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_qg(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9216_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_9217_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_9218_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_qh,N)),M) ).

% max_Suc2
tff(fact_9219_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_qi,N)),M) ).

% max_Suc1
tff(fact_9220_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,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2)))) ) ).

% card_le_Suc_Max
tff(fact_9221_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,aa_set_nat_nat(finite_card_nat,N4)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9222_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_ib),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ib),S2))) ).

% card_sum_le_nat_sum
tff(fact_9223_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_pn(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_9224_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_hb(N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_9225_diff__Suc,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(case_nat_nat(zero_zero_nat,aTP_Lamm_ib),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_9226_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aTP_Lamm_qj(X)) ).

% floor_real_def
tff(fact_9227_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ib),Nat) ).

% pred_def
tff(fact_9228_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9229_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,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9230_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_9231_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_9232_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_9233_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,aa_num_num(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_9234_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,aa_num_num(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_9235_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,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9236_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_9237_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(N),K) = aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% drop_bit_int_def
tff(fact_9238_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9239_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9240_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_9241_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)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9242_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_9243_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,aa_num_num(bit0,one))),N)) ).

% drop_bit_nat_def
tff(fact_9244_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
         => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_9245_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ? [R: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2))))
          & ! [N8: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N8),aa_set_nat_nat(finite_card_nat,S2)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R,N8)),S2)) ) ) ) ).

% finite_enumerate
tff(fact_9246_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9247_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa))))) ) ) ) ) ).

% bezw.elims
tff(fact_9248_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_9249_one__mod__minus__numeral,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_9250_minus__one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_9251_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_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(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_9252_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_mod_numeral
tff(fact_9253_Divides_Oadjust__mod__def,axiom,
    ! [R2: int,L: int] :
      ( ( ( R2 = zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R2) = zero_zero_int ) )
      & ( ( R2 != zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_9254_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa))))) ) ) )
           => ~ accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_9255_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa) = Y )
     => ( accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
               => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X))) ) ) )
           => ~ accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9256_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_qk(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_9257_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_9258_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,abs_abs_Code_integer(J)) ).

% divmod_abs_code(5)
tff(fact_9259_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc5135875973608410785r_bool(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K))) ).

% bit_cut_integer_def
tff(fact_9260_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_ql(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_qm(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_9261_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9262_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(aa_num_int(neg_numeral_sub_int(N),one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9263_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] : aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K),aa_num_int(neg_numeral_sub_int(N),one))) ).

% xor_minus_numerals(2)
tff(fact_9264_sub__BitM__One__eq,axiom,
    ! [N: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(N),one)) ).

% sub_BitM_One_eq
tff(fact_9265_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( ( L = zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
          & ( ( L != zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_qn(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_9266_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_ii,aTP_Lamm_hw) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9267_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_9268_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ~ pp(aa_set_nat_bool(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S2))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9269_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_9270_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_9271_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qp),Xa),X)) ).

% times_int.abs_eq
tff(fact_9272_int_Oabs__induct,axiom,
    ! [P: fun_int_bool,X: int] :
      ( ! [Y3: product_prod_nat_nat] : pp(aa_int_bool(P,aa_Pro4240683299535347697at_int(abs_Integ,Y3)))
     => pp(aa_int_bool(P,X)) ) ).

% int.abs_induct
tff(fact_9273_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)) ).

% eq_Abs_Integ
tff(fact_9274_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_9275_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9276_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_9277_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_qq),X)) ).

% uminus_int.abs_eq
tff(fact_9278_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9279_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qs),Xa),X)) ) ).

% less_int.abs_eq
tff(fact_9280_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qu),Xa),X)) ) ).

% less_eq_int.abs_eq
tff(fact_9281_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qw),Xa),X)) ).

% plus_int.abs_eq
tff(fact_9282_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qy),Xa),X)) ).

% minus_int.abs_eq
tff(fact_9283_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_9284_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M7,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9285_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,X),Y))),X) ).

% pow.simps(3)
tff(fact_9286_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q3)) = Q3 ).

% num_of_nat_numeral_eq
tff(fact_9287_sqr_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,N))) ).

% sqr.simps(2)
tff(fact_9288_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9289_sqr__conv__mult,axiom,
    ! [X: num] : aa_num_num(sqr,X) = aa_num_num(aa_num_fun_num_num(times_times_num,X),X) ).

% sqr_conv_mult
tff(fact_9290_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9291_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_9292_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_9293_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,X),Y)) ).

% pow.simps(2)
tff(fact_9294_num__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_num(code_num_of_integer,X) = aa_nat_num(num_of_nat,aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X))) ).

% num_of_integer.rep_eq
tff(fact_9295_num__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Code_integer_num(code_num_of_integer,aa_int_Code_integer(code_integer_of_int,X)) = aa_nat_num(num_of_nat,aa_int_nat(nat2,X)) ).

% num_of_integer.abs_eq
tff(fact_9296_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_9297_num__of__nat__plus__distrib,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,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9298_sqr_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,N)),N))) ).

% sqr.simps(3)
tff(fact_9299_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_9300_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qu),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa))) ) ).

% less_eq_int.rep_eq
tff(fact_9301_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qs),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa))) ) ).

% less_int.rep_eq
tff(fact_9302_Gcd__int__eq,axiom,
    ! [N4: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N4)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N4)) ).

% Gcd_int_eq
tff(fact_9303_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_qz,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9304_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5))) ).

% Gcd_int_greater_eq_0
tff(fact_9305_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).

% nat.rep_eq
tff(fact_9306_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_ra,M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9307_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_qq)) ).

% uminus_int_def
tff(fact_9308_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5)) ) ).

% Inf_nat_def1
tff(fact_9309_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_qp)) ).

% times_int_def
tff(fact_9310_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_qy)) ).

% minus_int_def
tff(fact_9311_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_qw)) ).

% plus_int_def
tff(fact_9312_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9313_Inf__real__def,axiom,
    ! [X7: set_real] : comple4887499456419720421f_real(X7) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X7))) ).

% Inf_real_def
tff(fact_9314_Inf__int__def,axiom,
    ! [X7: set_int] : complete_Inf_Inf_int(X7) = aa_int_int(uminus_uminus_int,complete_Sup_Sup_int(image_int_int(uminus_uminus_int,X7))) ).

% Inf_int_def
tff(fact_9315_suminf__eq__SUP__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X7,I3)))
       => ( suminf_real(X7) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_rb(X7),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9316_Sup__nat__def,axiom,
    ! [X7: set_nat] :
      ( ( ( X7 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X7) = zero_zero_nat ) )
      & ( ( X7 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X7) ) ) ) ).

% Sup_nat_def
tff(fact_9317_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_rc(N),K))) ).

% binomial_def
tff(fact_9318_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_rd,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9319_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_9320_card__UNIV__bool,axiom,
    finite_card_bool(top_top_set_bool) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9321_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_9322_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_9323_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_9324_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_9325_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9326_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,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9327_char_Osize_I2_J,axiom,
    ! [X15: bool,X2: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_size_char(char2(X15,X2,X32,X42,X52,X62,X72,X8)) = zero_zero_nat ).

% char.size(2)
tff(fact_9328_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,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% nat_of_char_less_256
tff(fact_9329_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,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9330_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B7: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B7)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(B7)),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B62))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B52))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B42))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B32))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B22))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B1))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B0)) ).

% integer_of_char_code
tff(fact_9331_char_Osize__gen,axiom,
    ! [X15: bool,X2: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_char(char2(X15,X2,X32,X42,X52,X62,X72,X8)) = zero_zero_nat ).

% char.size_gen
tff(fact_9332_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_9333_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,aa_num_num(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_9334_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D5: 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,aa_num_num(bit0,one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D5 = 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,aa_num_num(bit0,one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D5 = 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,aa_num_num(bit0,one))),N))
               => ( D5 = 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),D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9335_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_rf,suminf_real(aa_real_fun_nat_real(aTP_Lamm_rg,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9336_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => 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),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9337_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9338_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => 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),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9339_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9340_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)))
         => ( ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(A,B)))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9341_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,X: real,Y: real] :
      ( ! [X4: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real))
     => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9342_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9343_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => 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),H3)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9344_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9345_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))
       => ? [D6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D6))
                 => 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),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9346_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))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9347_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))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9348_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_or1222579329274155063t_real(A,B)))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1222579329274155063t_real(A,B)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X4))) )
       => ( 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_9349_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))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9350_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))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9351_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),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(F4,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9352_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D3: 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),D3))
       => ( ! [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))),D3))
             => ( aa_real_real(F,X) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9353_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_9354_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X4,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)),aa_num_real(numeral_numeral_real,aa_num_num(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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9355_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D3: 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),D3))
       => ( ! [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))),D3))
             => 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_9356_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D3: 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),D3))
       => ( ! [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))),D3))
             => 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_9357_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_9358_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rh,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_9359_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_ri(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_9360_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_rj,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_9361_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: 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_rk(F),N2),aa_nat_real(aa_real_fun_nat_real(F4,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(A,B)))
           => summable_real(aa_real_fun_nat_real(F,X4)) )
       => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(A,B)))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L5)
             => ( ! [N2: nat,X4: real,Y3: real] :
                    ( pp(aa_set_real_bool(member_real(X4),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,X4),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,X4),Y3))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_rl(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9362_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_rm(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_9363_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_9364_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_rn(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_9365_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))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9366_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,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9367_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_9368_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D5: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D5 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D5 = 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)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9369_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9370_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_9371_DERIV__power__series_H,axiom,
    ! [R3: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_ro(F),X4)) )
     => ( 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_rq(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ro(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9372_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_9373_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,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9374_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,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9375_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 )
        & ! [M4: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rr(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),T2)),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_9376_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 )
     => ( ! [M4: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rr(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),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9377_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,aa_num_num(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_9378_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 )
         => ( ! [M4: nat,T2: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rs(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),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9379_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 )
       => ( ! [M4: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ? [T2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rs(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),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9380_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 )
         => ( ! [M4: nat,T2: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rs(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),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9381_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 )
         => ( ! [M4: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rr(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),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9382_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 )
     => ( ! [M4: nat,T2: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rr(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),T2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9383_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 )
       => ( ! [M4: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,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))
             => ? [T2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),C))
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rt(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),T2)),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_9384_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 )
       => ( ! [M4: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,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))
             => ? [T2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),B))
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rt(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),T2)),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_9385_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 )
       => ( ! [M4: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,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 )
                   => ? [T2: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T2))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T2))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ru(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),T2)),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_9386_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B4: real] :
      ( ! [M4: nat,T2: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M4),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T5: 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),T5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_rw(N),Diff),B4),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rx(Diff),M2),T5)),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,B4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T5),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(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9387_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ? [L6: real,M8: real] :
            ( ! [X5: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),B)) )
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_real_real(F,X5)))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X5)),M8)) ) )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),M8)) )
               => ? [X4: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
                    & ( aa_real_real(F,X4) = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9388_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))
            & ! [X5: real] :
                ( ( ( X5 != 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),X5))),R)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X5))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9389_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))
            & ! [X5: real] :
                ( ( ( X5 != 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),X5))),R)) )
               => ( aa_real_real(F,X5) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9390_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))
            & ! [X5: real] :
                ( ( ( X5 != 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),X5))),R)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9391_isCont__real__sqrt,axiom,
    ! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9392_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_9393_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))
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9394_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_9395_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_9396_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_ry,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9397_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D5: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D5,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D5 != 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,D5),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9398_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_9399_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_9400_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),set_or1633881224788618240n_real(B,X)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( 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_9401_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_9402_isCont__inverse__function,axiom,
    ! [D3: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
     => ( ! [Z2: 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,Z2),X))),D3))
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: 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,Z2),X))),D3))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9403_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [Z2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z2),topolo2177554685111907308n_real(Z2,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(F4,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9404_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)))
         => ! [N8: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_rz(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N8))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N8)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9405_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))
         => ! [N8: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_rz(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N8)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N8)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9406_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_rz(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9407_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_9408_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_sa,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9409_monoseq__convergent,axiom,
    ! [X7: fun_nat_real,B4: real] :
      ( topolo6980174941875973593q_real(X7)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X7,I3))),B4))
       => ~ ! [L6: real] : ~ filterlim_nat_real(X7,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9410_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_sb,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9411_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_sc(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N8)),L3))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G,N8)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9412_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R: real] :
        ? [N6: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R),aa_nat_real(X7,N2))) )
     => filterlim_nat_real(aTP_Lamm_sd(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9413_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_se,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9414_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_sf,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9415_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_sg,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9416_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sh,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9417_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))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N8: 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,N8)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9418_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_9419_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_si(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9420_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_9421_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_9422_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_sj,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9423_LIMSEQ__inverse__real__of__nat__add__minus,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_minus
tff(fact_9424_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sl,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9425_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9426_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_rz(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9427_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_rz(A)) ) ) ) ).

% summable
tff(fact_9428_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_so(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9429_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_sp(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_so(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9430_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_sq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_rz(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9431_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_al,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9432_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_sq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_rz(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9433_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,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_rz(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9434_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] :
              ( ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N8)))),L3))
              & filterlim_nat_real(aTP_Lamm_sq(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N8: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),N8)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_sr(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9435_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_sr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_rz(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9436_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_sr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_rz(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9437_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ss,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_9438_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_9439_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_9440_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_9441_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_9442_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_9443_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_9444_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9445_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F3: 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,F3)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_st(N),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9446_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_9447_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F3: 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,F3)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_st(N),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9448_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_su,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9449_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_9450_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% quotient_of_number(3)
tff(fact_9451_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_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_9452_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9453_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q3: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q3)) ) ).

% quotient_of_denom_pos
tff(fact_9454_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_9455_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2)))) ).

% quotient_of_denom_pos'
tff(fact_9456_filterlim__real__at__infinity__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_infinity_real,at_top_nat) ).

% filterlim_real_at_infinity_sequentially
tff(fact_9457_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),Q3))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_sw(Q3)),quotient_of(P3))) ) ).

% rat_less_eq_code
tff(fact_9458_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q3))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_sy(Q3)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_9459_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_sz,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9460_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_9461_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_9462_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_ta,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9463_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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_9464_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_9465_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X4))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9466_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9467_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9468_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))
           => ! [T2: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T2))
               => ( R2 != aa_rat_rat(plus_plus_rat(S3),T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9469_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9470_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_rat_def
tff(fact_9471_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_te(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9472_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_tf(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9473_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_tf(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9474_eventually__at__top__to__right,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_tg(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_9475_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_tg(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_9476_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aTP_Lamm_tb(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),F3,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_th(F),G),F3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9477_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9478_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_tb(G2),at_top_real)
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),at_top_real)
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9479_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aTP_Lamm_tb(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_th(F),G),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9480_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_tb(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F0),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_th(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9481_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aTP_Lamm_tb(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_tb(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_th(F),G),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9482_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: 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_tb(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(F),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G2),F4),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9483_filterlim__int__sequentially,axiom,
    filterlim_nat_int(semiri1314217659103216013at_int,at_top_int,at_top_nat) ).

% filterlim_int_sequentially
tff(fact_9484_filterlim__nat__sequentially,axiom,
    filterlim_int_nat(nat2,at_top_nat,at_top_int) ).

% filterlim_nat_sequentially
tff(fact_9485_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X4))
         => pp(aa_nat_bool(P,X4)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9486_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N3))
         => pp(aa_nat_bool(P,N3)) ) ) ).

% eventually_sequentially
tff(fact_9487_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_9488_decseq__bounded,axiom,
    ! [X7: fun_nat_real,B4: real] :
      ( order_9091379641038594480t_real(X7)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),aa_nat_real(X7,I3)))
       => bfun_nat_real(X7,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_9489_decseq__convergent,axiom,
    ! [X7: fun_nat_real,B4: real] :
      ( order_9091379641038594480t_real(X7)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),aa_nat_real(X7,I3)))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X7,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_nat_real(X7,I2))) ) ) ) ).

% decseq_convergent
tff(fact_9490_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_9491_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9492_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B: nat] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_9493_Greatest__le__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)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_9494_GreatestI__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)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_9495_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_9496_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_9497_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))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9498_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9499_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))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,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_9500_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9501_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9502_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q3),zero_zero_int))
     => ( normalize(product_Pair_int_int(P3,Q3)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q3))) ) ) ).

% normalize_negative
tff(fact_9503_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)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L3: real,Z2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z2,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_9504_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_9505_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),A3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_ti(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9506_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ? [C2: real,D6: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C2,D6) )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D6)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_9507_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: 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)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                & ! [X5: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z2),X5) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9508_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Xi))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi),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(F4,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9509_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q3: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q3) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q3)) ) ).

% normalize_denom_pos
tff(fact_9510_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P3: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P3,Q3)) = normalize(product_Pair_int_int(R2,S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9511_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))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9512_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))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,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_9513_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)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9514_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)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,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_9515_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)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),B))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9516_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( normalize(P3) = product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))),aa_int_int(divide_divide_int(product_snd_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( ( ( product_snd_int_int(P3) = zero_zero_int )
           => ( normalize(P3) = product_Pair_int_int(zero_zero_int,one_one_int) ) )
          & ( ( product_snd_int_int(P3) != zero_zero_int )
           => ( normalize(P3) = product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))),aa_int_int(divide_divide_int(product_snd_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_9517_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9518_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2_int
tff(fact_9519_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_9520_gcd__0__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = aa_int_int(abs_abs_int,X) ).

% gcd_0_int
tff(fact_9521_gcd__0__left__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = aa_int_int(abs_abs_int,X) ).

% gcd_0_left_int
tff(fact_9522_gcd__ge__0__int,axiom,
    ! [X: int,Y: 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(gcd_gcd_int,X),Y))) ).

% gcd_ge_0_int
tff(fact_9523_gcd__le1__int,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,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A)) ) ).

% gcd_le1_int
tff(fact_9524_gcd__le2__int,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,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B)) ) ).

% gcd_le2_int
tff(fact_9525_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_bool] :
      ( ( 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(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),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,Y),zero_zero_int))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
       => ( ( 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,zero_zero_int),Y))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
         => ( ( 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))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_9526_gcd__unique__int,axiom,
    ! [D3: int,A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D3))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D3),B))
        & ! [E3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),A))
              & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),B)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),D3)) ) )
    <=> ( D3 = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9527_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(modulo_modulo_int(X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9528_gcd__code__int,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,L),zero_zero_int),K),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9529_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_tj(N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9530_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9531_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9532_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_9533_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9534_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_9535_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_9536_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_9537_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9538_gcd__pos__nat,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(gcd_gcd_nat,M),N)))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9539_gcd__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) ).

% gcd_int_int_eq
tff(fact_9540_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_9541_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% gcd_nat_abs_left_eq
tff(fact_9542_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9543_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_9544_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_9545_gcd__diff1__nat,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(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff1_nat
tff(fact_9546_gcd__diff2__nat,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(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff2_nat
tff(fact_9547_gcd__le2__nat,axiom,
    ! [B: nat,A: 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(gcd_gcd_nat,A),B)),B)) ) ).

% gcd_le2_nat
tff(fact_9548_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).

% gcd_le1_nat
tff(fact_9549_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9550_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] : gcd_gcd_Code_integer(K,L) = abs_abs_Code_integer(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,L),zero_z3403309356797280102nteger),K,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9551_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_9552_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9553_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y3: 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),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),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),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9554_incseq__bounded,axiom,
    ! [X7: fun_nat_real,B4: real] :
      ( order_mono_nat_real(X7)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X7,I3)),B4))
       => bfun_nat_real(X7,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9555_incseq__convergent,axiom,
    ! [X7: fun_nat_real,B4: real] :
      ( order_mono_nat_real(X7)
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X7,I3)),B4))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X7,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X7,I2)),L6)) ) ) ) ).

% incseq_convergent
tff(fact_9556_gcd__int__def,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).

% gcd_int_def
tff(fact_9557_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_tk) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9558_gcd__is__Max__divisors__nat,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(gcd_gcd_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tl(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9559_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9560_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9561_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y )
     => ( accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)) ) ) )
           => ~ accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9562_complex__is__Nat__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_set_complex_bool(member_complex(Z),semiri3842193898606819883omplex))
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I4: nat] : aa_complex_real(re,Z) = aa_nat_real(semiri5074537144036343181t_real,I4) ) ) ).

% complex_is_Nat_iff
tff(fact_9563_tendsto__at__topI__sequentially__real,axiom,
    ! [F: fun_real_real,Y: real] :
      ( order_mono_real_real(F)
     => ( filterlim_nat_real(aTP_Lamm_tn(F),topolo2815343760600316023s_real(Y),at_top_nat)
       => filterlim_real_real(F,topolo2815343760600316023s_real(Y),at_top_real) ) ) ).

% tendsto_at_topI_sequentially_real
tff(fact_9564_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_to(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9565_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_9566_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_9567_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_9568_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_qc,K)),N4)) ) ).

% inj_on_diff_nat
tff(fact_9569_inj__Suc,axiom,
    ! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).

% inj_Suc
tff(fact_9570_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9571_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F))) ) ) ) ).

% suminf_reindex_mono
tff(fact_9572_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( ( X != zero_zero_real )
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N)) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X),N) ) ) ) ).

% powr_real_of_int'
tff(fact_9573_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,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_9574_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
         => ( ! [X4: nat] :
                ( ~ pp(aa_set_nat_bool(member_nat(X4),image_nat_nat(G,top_top_set_nat)))
               => ( aa_nat_real(F,X4) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9575_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F4: fun_real_real] :
      ( ! [X4: real] : has_fi5821293074295781190e_real(F,aa_real_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real))
     => ( ! [X4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F4,X4)))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9576_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa_num_num(bit0,one)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9577_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9578_integer__of__num_I2_J,axiom,
    ! [N: num] : code_integer_of_num(aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(code_integer_of_num(N),code_integer_of_num(N)) ).

% integer_of_num(2)
tff(fact_9579_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9580_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9581_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_9582_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9583_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa))
      <=> pp(Y) )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( pp(Y)
          <=> ( Xa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9584_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa))
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xa )
                  & ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                  & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9585_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg3))
    <=> ( ( Deg = Deg3 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg),TreeList),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9586_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa))
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xa )
                & ! [X4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9587_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( pp(Y)
                <=> ( Xa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg2),TreeList2),Summary2))),Mima)) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9588_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa))
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa))
                 => ~ ( ( Deg2 = Xa )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X5),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9589_Sup__int__def,axiom,
    ! [X7: set_int] : complete_Sup_Sup_int(X7) = the_int(aTP_Lamm_tq(X7)) ).

% Sup_int_def
tff(fact_9590_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa))
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa))
                 => ( ( Deg2 = Xa )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9591_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_tr(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9592_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_9593_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_9594_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_9595_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ts,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_9596_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_9597_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ts,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M)) ).

% take_bit_num_simps(6)
tff(fact_9598_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_9599_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_tt(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9600_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_tu,N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9601_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_tv(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9602_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_9603_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,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).

% and_minus_numerals(3)
tff(fact_9604_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,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).

% and_minus_numerals(7)
tff(fact_9605_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,aa_num_num(bit0,N))) ).

% and_minus_numerals(4)
tff(fact_9606_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,aa_num_num(bit0,N))) ).

% and_minus_numerals(8)
tff(fact_9607_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9608_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit0,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(4)
tff(fact_9609_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one,aa_num_num(bit0,N)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9610_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_9611_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(aa_num_num(bit0,M)) ).

% and_not_num.simps(7)
tff(fact_9612_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q3: num] :
      ( ( bit_and_not_num(M,N) = some_num(Q3) )
    <=> ( 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,Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9613_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tw,bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_9614_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_9615_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_9616_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_9617_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_ua,product_Pair_nat_num(N,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9618_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_ub) ).

% Rats_eq_int_div_nat
tff(fact_9619_Rats__abs__iff,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(aa_real_real(abs_abs_real,X)),field_5140801741446780682s_real))
    <=> pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real)) ) ).

% Rats_abs_iff
tff(fact_9620_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9621_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X)) ) ).

% Rats_no_bot_less
tff(fact_9622_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( pp(aa_set_real_bool(member_real(X4),field_5140801741446780682s_real))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X4)) ) ).

% Rats_no_top_le
tff(fact_9623_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_uc) ).

% Rats_eq_int_div_int
tff(fact_9624_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] : Xa = aa_num_num(bit0,N2)
             => ( Y != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] : Xa = aa_num_num(bit1,N2)
               => ( Y != none_num ) ) )
           => ( ! [M4: num] :
                  ( ( X = aa_num_num(bit0,M4) )
                 => ( ( Xa = one )
                   => ( Y != some_num(aa_num_num(bit0,M4)) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M4,N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M4,N2)) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit1,M4) )
                       => ( ( Xa = one )
                         => ( Y != some_num(aa_num_num(bit0,M4)) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa_num_num(bit0,N2) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tw,bit_and_not_num(M4,N2)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M4,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9625_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(6)
tff(fact_9626_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(9)
tff(fact_9627_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(5)
tff(fact_9628_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit0,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(5)
tff(fact_9629_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(6)
tff(fact_9630_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(9)
tff(fact_9631_xor__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),one) = none_num ).

% xor_num.simps(1)
tff(fact_9632_xor__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,X),Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N2: num] :
                ( ( Xa = aa_num_num(bit0,N2) )
               => ( Y != some_num(aa_num_num(bit1,N2)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa = aa_num_num(bit1,N2) )
                 => ( Y != some_num(aa_num_num(bit0,N2)) ) ) )
           => ( ! [M4: num] :
                  ( ( X = aa_num_num(bit0,M4) )
                 => ( ( Xa = one )
                   => ( Y != some_num(aa_num_num(bit1,M4)) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit1,N2) )
                         => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit1,M4) )
                       => ( ( Xa = one )
                         => ( Y != some_num(aa_num_num(bit0,M4)) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa_num_num(bit0,N2) )
                             => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2))) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9633_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),one) = some_num(aa_num_num(bit0,M)) ).

% xor_num.simps(7)
tff(fact_9634_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),one) = some_num(aa_num_num(bit1,M)) ).

% xor_num.simps(4)
tff(fact_9635_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,N)) = some_num(aa_num_num(bit0,N)) ).

% xor_num.simps(3)
tff(fact_9636_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,N)) = some_num(aa_num_num(bit1,N)) ).

% xor_num.simps(2)
tff(fact_9637_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(8)
tff(fact_9638_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,X),Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] : Xa = aa_num_num(bit0,N2)
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] : Xa = aa_num_num(bit1,N2)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M4: num] : X = aa_num_num(bit0,M4)
               => ( ( Xa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) ) ) )
                 => ( ( ? [M4: num] : X = aa_num_num(bit1,M4)
                     => ( ( Xa = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa_num_num(bit0,N2) )
                             => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit1,N2) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tw,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9639_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_9640_and__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),one) = some_num(one) ).

% and_num.simps(1)
tff(fact_9641_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(5)
tff(fact_9642_and__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9643_and__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,N)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9644_and__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,N)) = none_num ).

% and_num.simps(2)
tff(fact_9645_and__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),one) = none_num ).

% and_num.simps(4)
tff(fact_9646_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(8)
tff(fact_9647_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(6)
tff(fact_9648_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tw,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(9)
tff(fact_9649_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_9650_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9651_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_qu)) ).

% less_eq_int_def
tff(fact_9652_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_qs)) ).

% less_int_def
tff(fact_9653_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9654_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_9655_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_9656_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_9657_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q3) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9658_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q3),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9659_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_9660_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_9661_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_9662_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_9663_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: 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),Q3)) = 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),Q3)) ).

% nat_mult_min_right
tff(fact_9664_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: 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)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ).

% nat_mult_min_left
tff(fact_9665_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_9666_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ud,N)),M) ).

% min_Suc2
tff(fact_9667_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ue,N)),M) ).

% min_Suc1
tff(fact_9668_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9669_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9670_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_9671_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: 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,J),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),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_9672_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa = aa_num_num(bit0,N2) )
                 => ( ( Y = some_num(one) )
                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa = aa_num_num(bit1,N2) )
                   => ( ( Y = none_num )
                     => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ( ( Xa = one )
                     => ( ( Y = some_num(aa_num_num(bit0,M4)) )
                       => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M4),one)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M4,N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa_num_num(bit1,N2) )
                           => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M4,N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ( ( Xa = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M4)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M4),one)) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit0,N2) )
                               => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tw,bit_and_not_num(M4,N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa_num_num(bit1,N2) )
                                 => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M4,N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9673_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: 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,J),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),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_9674_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,X),Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = some_num(one) )
               => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa = aa_num_num(bit0,N2) )
                 => ( ( Y = none_num )
                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa = aa_num_num(bit1,N2) )
                   => ( ( Y = some_num(one) )
                     => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M4),one)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa_num_num(bit1,N2) )
                           => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ( ( Xa = one )
                           => ( ( Y = some_num(one) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M4),one)) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit0,N2) )
                               => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa_num_num(bit1,N2) )
                                 => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tw,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9675_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,X),Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa = aa_num_num(bit0,N2) )
                 => ( ( Y = some_num(aa_num_num(bit1,N2)) )
                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa = aa_num_num(bit1,N2) )
                   => ( ( Y = some_num(aa_num_num(bit0,N2)) )
                     => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ( ( Xa = one )
                     => ( ( Y = some_num(aa_num_num(bit1,M4)) )
                       => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M4),one)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa_num_num(bit1,N2) )
                           => ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2))) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ( ( Xa = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M4)) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M4),one)) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit0,N2) )
                               => ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2))) )
                                 => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa_num_num(bit1,N2) )
                                 => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_9676_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X),Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa = aa_num_num(bit0,M4) )
                 => ( ( Y = aa_num_num(bit1,M4) )
                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit0,M4))) ) ) )
           => ( ( ( X = one )
               => ! [M4: num] :
                    ( ( Xa = aa_num_num(bit1,M4) )
                   => ( ( Y = aa_num_num(bit1,M4) )
                     => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit1,M4))) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa_num_num(bit0,N2) )
                   => ( ( Xa = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N2),one)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa_num_num(bit0,N2) )
                     => ! [M4: num] :
                          ( ( Xa = aa_num_num(bit0,M4) )
                         => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) )
                           => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit0,M4))) ) ) )
                 => ( ! [N2: num] :
                        ( ( X = aa_num_num(bit0,N2) )
                       => ! [M4: num] :
                            ( ( Xa = aa_num_num(bit1,M4) )
                           => ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit1,M4))) ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa_num_num(bit1,N2) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N2),one)) ) ) )
                     => ( ! [N2: num] :
                            ( ( X = aa_num_num(bit1,N2) )
                           => ! [M4: num] :
                                ( ( Xa = aa_num_num(bit0,M4) )
                               => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) )
                                 => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit0,M4))) ) ) )
                       => ~ ! [N2: num] :
                              ( ( X = aa_num_num(bit1,N2) )
                             => ! [M4: num] :
                                  ( ( Xa = aa_num_num(bit1,M4) )
                                 => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M4)) )
                                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit1,M4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9677_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_9678_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_9679_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_uf(M),N4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ug(M),N4)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_uh(M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9680_Code__Target__Nat_ONat_Oabs__eq,axiom,
    ! [X: int] : aa_Code_integer_nat(code_Target_Nat,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(nat2,X) ).

% Code_Target_Nat.Nat.abs_eq
tff(fact_9681_Code__Target__Nat_ONat_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_nat(code_Target_Nat,X) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% Code_Target_Nat.Nat.rep_eq
tff(fact_9682_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9683_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9684_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_9685_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_9686_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
               => ( Y = nil_int ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa)) ) ) ) ).

% upto.pelims
tff(fact_9687_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_9688_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_9689_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9690_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9691_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9692_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(aa_int_fun_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_9693_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(aa_int_fun_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_9694_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(aa_int_fun_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_9695_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(aa_int_fun_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_9696_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9697_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9698_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_9699_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_9700_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9701_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9702_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9703_le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ).

% le_enumerate
tff(fact_9704_finite__le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ) ).

% finite_le_enumerate
tff(fact_9705_DeMoivre2,axiom,
    ! [R2: real,A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(rcis(R2),A)),N) = aa_real_complex(rcis(aa_nat_real(aa_real_fun_nat_real(power_power_real,R2),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).

% DeMoivre2
tff(fact_9706_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9707_rcis__zero__arg,axiom,
    ! [R2: real] : aa_real_complex(rcis(R2),zero_zero_real) = aa_real_complex(real_V4546457046886955230omplex,R2) ).

% rcis_zero_arg
tff(fact_9708_rcis__zero__mod,axiom,
    ! [A: real] : aa_real_complex(rcis(zero_zero_real),A) = zero_zero_complex ).

% rcis_zero_mod
tff(fact_9709_rcis__eq__zero__iff,axiom,
    ! [R2: real,A: real] :
      ( ( aa_real_complex(rcis(R2),A) = zero_zero_complex )
    <=> ( R2 = zero_zero_real ) ) ).

% rcis_eq_zero_iff
tff(fact_9710_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K2: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K2)))
              <=> pp(aa_nat_bool(Q,K2)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9711_Least__Suc,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_ui(P))) ) ) ) ).

% Least_Suc
tff(fact_9712_Inf__nat__def,axiom,
    ! [X7: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,X7) = ord_Least_nat(aTP_Lamm_ad(X7)) ).

% Inf_nat_def
tff(fact_9713_Sup__real__def,axiom,
    ! [X7: set_real] : comple1385675409528146559p_real(X7) = ord_Least_real(aTP_Lamm_uj(X7)) ).

% Sup_real_def
tff(fact_9714_Code__Target__Nat_ONat__def,axiom,
    code_Target_Nat = map_fu6539832666145259331at_nat(code_int_of_integer,id_nat,nat2) ).

% Code_Target_Nat.Nat_def
tff(fact_9715_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ib),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).

% sum_list_upt
tff(fact_9716_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9717_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9718_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9719_take__upt,axiom,
    ! [I: 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,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9720_nth__upt,axiom,
    ! [I: nat,K: nat,J: 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)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9721_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_9722_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9723_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_9724_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9725_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_9726_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9727_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9728_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9729_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9730_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9731_nat__of__integer__def,axiom,
    code_nat_of_integer = map_fu6539832666145259331at_nat(code_int_of_integer,id_nat,nat2) ).

% nat_of_integer_def
tff(fact_9732_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_9733_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_9734_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uk,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_9735_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_ul,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9736_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9737_sorted__upto,axiom,
    ! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).

% sorted_upto
tff(fact_9738_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).

% sorted_wrt_upto
tff(fact_9739_num__of__integer__def,axiom,
    code_num_of_integer = map_fu1227494855608507351um_num(code_int_of_integer,id_num,comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer_def
tff(fact_9740_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9741_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_un,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9742_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_up,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9743_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(zero_zero_int,one_one_int)) ).

% zero_rat_def
tff(fact_9744_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(X)),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(product_snd_int_int(X),product_fst_int_int(X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_9745_ratrel__iff,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),Y))
    <=> ( ( product_snd_int_int(X) != zero_zero_int )
        & ( product_snd_int_int(Y) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Y)),product_snd_int_int(X)) ) ) ) ).

% ratrel_iff
tff(fact_9746_zero__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,product_Pair_int_int(zero_zero_int,one_one_int)),product_Pair_int_int(zero_zero_int,one_one_int))) ).

% zero_rat.rsp
tff(fact_9747_ratrel__def,axiom,
    ! [X5: product_prod_int_int,Xa2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X5),Xa2))
    <=> ( ( product_snd_int_int(X5) != zero_zero_int )
        & ( product_snd_int_int(Xa2) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X5)),product_snd_int_int(Xa2)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa2)),product_snd_int_int(X5)) ) ) ) ).

% ratrel_def
tff(fact_9748_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( pp(aa_rat_bool(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_9749_Rat_Opositive__zero,axiom,
    ~ pp(aa_rat_bool(positive,zero_zero_rat)) ).

% Rat.positive_zero
tff(fact_9750_Rat_Opositive__minus,axiom,
    ! [X: rat] :
      ( ~ pp(aa_rat_bool(positive,X))
     => ( ( X != zero_zero_rat )
       => pp(aa_rat_bool(positive,aa_rat_rat(uminus_uminus_rat,X))) ) ) ).

% Rat.positive_minus
tff(fact_9751_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(positive,X))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_9752_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_uq) ).

% Rat.positive_def
tff(fact_9753_inverse__rat__def,axiom,
    inverse_inverse_rat = map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat,aTP_Lamm_ur) ).

% inverse_rat_def
tff(fact_9754_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Q4: rat] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,Q4)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q4)),Y)) ) ) ).

% of_rat_dense
tff(fact_9755_Rat_Opositive_Orsp,axiom,
    bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_uq,aTP_Lamm_uq) ).

% Rat.positive.rsp
tff(fact_9756_inverse__rat_Orsp,axiom,
    bNF_re7145576690424134365nt_int(ratrel,ratrel,aTP_Lamm_ur,aTP_Lamm_ur) ).

% inverse_rat.rsp
tff(fact_9757_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_us,aTP_Lamm_us) ).

% Fract.rsp
tff(fact_9758_integer__of__natural_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,semiri1314217659103216013at_int,semiri1314217659103216013at_int) ).

% integer_of_natural.rsp
tff(fact_9759_natural__of__integer_Orsp,axiom,
    bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,nat2,nat2) ).

% natural_of_integer.rsp
tff(fact_9760_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_ut,aTP_Lamm_ut) ).

% sub.rsp
tff(fact_9761_less__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9762_less__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9763_less__eq__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_9764_less__eq__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9765_num__of__integer_Orsp,axiom,
    bNF_re7626690874201225453um_num(fequal_int,fequal_num,comp_nat_num_int(num_of_nat,nat2),comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer.rsp
tff(fact_9766_inverse__rat_Otransfer,axiom,
    bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat,aTP_Lamm_ur,inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_9767_Rat_Opositive_Otransfer,axiom,
    bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_uq,positive) ).

% Rat.positive.transfer
tff(fact_9768_zero__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,product_Pair_int_int(zero_zero_int,one_one_int)),zero_zero_rat)) ).

% zero_rat.transfer
tff(fact_9769_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_qp),times_times_int) ).

% times_int.transfer
tff(fact_9770_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_9771_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_uu,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9772_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_qq)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_9773_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9774_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_9775_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_qs),ord_less_int) ).

% less_int.transfer
tff(fact_9776_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_qu),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9777_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_qw),plus_plus_int) ).

% plus_int.transfer
tff(fact_9778_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_qy),minus_minus_int) ).

% minus_int.transfer
tff(fact_9779_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_qp),produc4528391901023682385at_nat(aTP_Lamm_qp)) ).

% times_int.rsp
tff(fact_9780_Real_Opositive_Orsp,axiom,
    bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_uv,aTP_Lamm_uv) ).

% Real.positive.rsp
tff(fact_9781_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9782_plus__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_uw,aTP_Lamm_uw) ).

% plus_real.rsp
tff(fact_9783_int_Orel__eq__transfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_9784_times__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_ux,aTP_Lamm_ux) ).

% times_real.rsp
tff(fact_9785_uminus__real_Orsp,axiom,
    pp(aa_fun1696419067750587730t_bool(aa_fun4901480949278917796t_bool(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_uy),aTP_Lamm_uy)) ).

% uminus_real.rsp
tff(fact_9786_one__real_Orsp,axiom,
    pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_uz),aTP_Lamm_uz)) ).

% one_real.rsp
tff(fact_9787_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4240683299535347697at_int(abs_Integ,X) = aa_Pro4240683299535347697at_int(abs_Integ,Y) )
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,X),Y)) ) ).

% int.abs_eq_iff
tff(fact_9788_zero__real_Orsp,axiom,
    pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_va),aTP_Lamm_va)) ).

% zero_real.rsp
tff(fact_9789_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_9790_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_qq)),produc2626176000494625587at_nat(aTP_Lamm_qq))) ).

% uminus_int.rsp
tff(fact_9791_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_9792_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_9793_intrel__def,axiom,
    intrel = produc5023667602222594023t_bool(aTP_Lamm_vc) ).

% intrel_def
tff(fact_9794_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_qs),produc5023667602222594023t_bool(aTP_Lamm_qs)) ).

% less_int.rsp
tff(fact_9795_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_qu),produc5023667602222594023t_bool(aTP_Lamm_qu)) ).

% less_eq_int.rsp
tff(fact_9796_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_qw),produc4528391901023682385at_nat(aTP_Lamm_qw)) ).

% plus_int.rsp
tff(fact_9797_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_qy),produc4528391901023682385at_nat(aTP_Lamm_qy)) ).

% minus_int.rsp
tff(fact_9798_int_Obi__total,axiom,
    bi_tot896582865486249351at_int(pcr_int) ).

% int.bi_total
tff(fact_9799_inverse__real_Orsp,axiom,
    pp(aa_fun1696419067750587730t_bool(aa_fun4901480949278917796t_bool(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_ve),aTP_Lamm_ve)) ).

% inverse_real.rsp
tff(fact_9800_vanishes__mult__bounded,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( ? [A7: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A7))
          & ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N2))),A7)) )
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ux,X7),Y6))) ) ) ).

% vanishes_mult_bounded
tff(fact_9801_vanishes__const,axiom,
    ! [C: rat] :
      ( pp(vanishes(aTP_Lamm_vf(C)))
    <=> ( C = zero_zero_rat ) ) ).

% vanishes_const
tff(fact_9802_vanishes__add,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( pp(vanishes(X7))
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_uw,X7),Y6))) ) ) ).

% vanishes_add
tff(fact_9803_vanishes__diff,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( pp(vanishes(X7))
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vg(X7),Y6))) ) ) ).

% vanishes_diff
tff(fact_9804_vanishes__minus,axiom,
    ! [X7: fun_nat_rat] :
      ( pp(vanishes(X7))
     => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_uy,X7))) ) ).

% vanishes_minus
tff(fact_9805_vanishes__def,axiom,
    ! [X7: fun_nat_rat] :
      ( pp(vanishes(X7))
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N3))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9806_vanishesI,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [R: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
         => ? [K4: nat] :
            ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N2))),R)) ) )
     => pp(vanishes(X7)) ) ).

% vanishesI
tff(fact_9807_vanishesD,axiom,
    ! [X7: fun_nat_rat,R2: rat] :
      ( pp(vanishes(X7))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [N8: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N8))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N8))),R2)) ) ) ) ).

% vanishesD
tff(fact_9808_inverse__real_Otransfer,axiom,
    pp(aa_fun9137951878280015556l_bool(aa_fun9159459243913713430l_bool(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_ve),inverse_inverse_real)) ).

% inverse_real.transfer
tff(fact_9809_inverse__real_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,if_fun_nat_rat(vanishes(X),aTP_Lamm_va,aTP_Lamm_vd(X))) ) ) ).

% inverse_real.abs_eq
tff(fact_9810_real_Oabs__induct,axiom,
    ! [P: fun_real_bool,X: real] :
      ( ! [Y3: fun_nat_rat] :
          ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Y3),Y3))
         => pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,Y3))) )
     => pp(aa_real_bool(P,X)) ) ).

% real.abs_induct
tff(fact_9811_of__rat__Real,axiom,
    ! [X: rat] : aa_rat_real(field_7254667332652039916t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_vf(X)) ).

% of_rat_Real
tff(fact_9812_zero__real__def,axiom,
    zero_zero_real = aa_fun_nat_rat_real(real2,aTP_Lamm_va) ).

% zero_real_def
tff(fact_9813_one__real__def,axiom,
    one_one_real = aa_fun_nat_rat_real(real2,aTP_Lamm_uz) ).

% one_real_def
tff(fact_9814_real_Orel__eq__transfer,axiom,
    bNF_re8935275315890131625l_bool(pcr_real,bNF_re1458715823057739248l_bool(pcr_real,fequal_bool),realrel,fequal_real) ).

% real.rel_eq_transfer
tff(fact_9815_of__int__Real,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_vh(X)) ).

% of_int_Real
tff(fact_9816_zero__real_Otransfer,axiom,
    pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,aTP_Lamm_va),zero_zero_real)) ).

% zero_real.transfer
tff(fact_9817_one__real_Otransfer,axiom,
    pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,aTP_Lamm_uz),one_one_real)) ).

% one_real.transfer
tff(fact_9818_of__nat__Real,axiom,
    ! [X: nat] : aa_nat_real(semiri5074537144036343181t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_vi(X)) ).

% of_nat_Real
tff(fact_9819_uminus__real_Otransfer,axiom,
    pp(aa_fun9137951878280015556l_bool(aa_fun9159459243913713430l_bool(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_uy),uminus_uminus_real)) ).

% uminus_real.transfer
tff(fact_9820_plus__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_uw,plus_plus_real) ).

% plus_real.transfer
tff(fact_9821_uminus__real_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_uy,X)) ) ) ).

% uminus_real.abs_eq
tff(fact_9822_times__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_ux,times_times_real) ).

% times_real.transfer
tff(fact_9823_plus__real_Oabs__eq,axiom,
    ! [Xa: fun_nat_rat,X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa),Xa))
     => ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,Xa)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_uw,Xa),X)) ) ) ) ).

% plus_real.abs_eq
tff(fact_9824_times__real_Oabs__eq,axiom,
    ! [Xa: fun_nat_rat,X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa),Xa))
     => ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,Xa)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ux,Xa),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_9825_Real_Opositive_Otransfer,axiom,
    pp(aa_fun3352386287069783311l_bool(aa_fun7221085148252759141l_bool(bNF_re1458715823057739248l_bool(pcr_real,fequal_bool),aTP_Lamm_uv),positive2)) ).

% Real.positive.transfer
tff(fact_9826_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X,N3))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_9827_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(positive2,X))
     => ( pp(aa_real_bool(positive2,Y))
       => pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))) ) ) ).

% Real.positive_mult
tff(fact_9828_Real_Opositive__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(positive2,X))
     => ( pp(aa_real_bool(positive2,Y))
       => pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% Real.positive_add
tff(fact_9829_Real_Opositive__zero,axiom,
    ~ pp(aa_real_bool(positive2,zero_zero_real)) ).

% Real.positive_zero
tff(fact_9830_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(positive2,X))
     => ( ( X != zero_zero_real )
       => pp(aa_real_bool(positive2,aa_real_real(uminus_uminus_real,X))) ) ) ).

% Real.positive_minus
tff(fact_9831_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X))) ) ).

% less_real_def
tff(fact_9832_le__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X7,N3)),aa_rat_rat(plus_plus_rat(aa_nat_rat(Y6,N3)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9833_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X),N3))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9834_realrel__refl,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X7),X7)) ) ).

% realrel_refl
tff(fact_9835_cauchy__minus,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_uy,X7)) ) ).

% cauchy_minus
tff(fact_9836_cauchy__const,axiom,
    ! [X: rat] : cauchy(aTP_Lamm_vf(X)) ).

% cauchy_const
tff(fact_9837_cauchy__add,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_uw,X7),Y6)) ) ) ).

% cauchy_add
tff(fact_9838_cauchy__mult,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ux,X7),Y6)) ) ) ).

% cauchy_mult
tff(fact_9839_Real__induct,axiom,
    ! [P: fun_real_bool,X: real] :
      ( ! [X9: fun_nat_rat] :
          ( cauchy(X9)
         => pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,X9))) )
     => pp(aa_real_bool(P,X)) ) ).

% Real_induct
tff(fact_9840_cauchy__diff,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_vg(X7),Y6)) ) ) ).

% cauchy_diff
tff(fact_9841_cr__real__eq,axiom,
    ! [X5: fun_nat_rat,Xa2: real] :
      ( pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,X5),Xa2))
    <=> ( cauchy(X5)
        & ( aa_fun_nat_rat_real(real2,X5) = Xa2 ) ) ) ).

% cr_real_eq
tff(fact_9842_cauchy__inverse,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => cauchy(aTP_Lamm_vd(X7)) ) ) ).

% cauchy_inverse
tff(fact_9843_cauchy__imp__bounded,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ? [B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
          & ! [N8: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N8))),B3)) ) ) ).

% cauchy_imp_bounded
tff(fact_9844_less__RealD,axiom,
    ! [Y6: fun_nat_rat,X: real] :
      ( cauchy(Y6)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_fun_nat_rat_real(real2,Y6)))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N2)))) ) ) ).

% less_RealD
tff(fact_9845_le__RealI,axiom,
    ! [Y6: fun_nat_rat,X: real] :
      ( cauchy(Y6)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N2))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_fun_nat_rat_real(real2,Y6))) ) ) ).

% le_RealI
tff(fact_9846_Real__leI,axiom,
    ! [X7: fun_nat_rat,Y: real] :
      ( cauchy(X7)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X7,N2))),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X7)),Y)) ) ) ).

% Real_leI
tff(fact_9847_minus__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X7)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_uy,X7)) ) ) ).

% minus_Real
tff(fact_9848_add__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_uw,X7),Y6)) ) ) ) ).

% add_Real
tff(fact_9849_mult__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ux,X7),Y6)) ) ) ) ).

% mult_Real
tff(fact_9850_diff__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_fun_nat_rat_real(real2,X7)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_vg(X7),Y6)) ) ) ) ).

% diff_Real
tff(fact_9851_realrelI,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vg(X7),Y6)))
         => pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X7),Y6)) ) ) ) ).

% realrelI
tff(fact_9852_eq__Real,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( ( aa_fun_nat_rat_real(real2,X7) = aa_fun_nat_rat_real(real2,Y6) )
        <=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vg(X7),Y6))) ) ) ) ).

% eq_Real
tff(fact_9853_vanishes__diff__inverse,axiom,
    ! [X7: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ( cauchy(Y6)
         => ( ~ pp(vanishes(Y6))
           => ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vg(X7),Y6)))
             => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vj(X7),Y6))) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_9854_realrel__def,axiom,
    ! [X5: fun_nat_rat,Xa2: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X5),Xa2))
    <=> ( cauchy(X5)
        & cauchy(Xa2)
        & pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_vg(X5),Xa2))) ) ) ).

% realrel_def
tff(fact_9855_cauchy__not__vanishes__cases,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
                ( ! [N8: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N8))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X7,N8)))) )
                | ! [N8: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N8))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_nat_rat(X7,N8))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9856_positive__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X7)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X7,N3))) ) ) ) ) ).

% positive_Real
tff(fact_9857_cauchy__not__vanishes,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
              ! [N8: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N8))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X7,N8)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9858_cauchy__def,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M3))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X7,M3)),aa_nat_rat(X7,N3)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9859_cauchyI,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [R: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
         => ? [K4: nat] :
            ! [M4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M4))
             => ! [N2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X7,M4)),aa_nat_rat(X7,N2)))),R)) ) ) )
     => cauchy(X7) ) ).

% cauchyI
tff(fact_9860_cauchyD,axiom,
    ! [X7: fun_nat_rat,R2: rat] :
      ( cauchy(X7)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
           => ! [N8: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N8))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X7,M2)),aa_nat_rat(X7,N8)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_9861_inverse__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ( pp(vanishes(X7))
         => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X7)) = zero_zero_real ) )
        & ( ~ pp(vanishes(X7))
         => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X7)) = aa_fun_nat_rat_real(real2,aTP_Lamm_vd(X7)) ) ) ) ) ).

% inverse_Real
tff(fact_9862_not__positive__Real,axiom,
    ! [X7: fun_nat_rat] :
      ( cauchy(X7)
     => ( ~ pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X7)))
      <=> ! [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
           => ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X7,N3)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9863_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_uv) ).

% Real.positive_def
tff(fact_9864_inverse__real__def,axiom,
    inverse_inverse_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_ve) ).

% inverse_real_def
tff(fact_9865_uminus__real__def,axiom,
    uminus_uminus_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_uy) ).

% uminus_real_def
tff(fact_9866_times__real__def,axiom,
    times_times_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_ux) ).

% times_real_def
tff(fact_9867_plus__real__def,axiom,
    plus_plus_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_uw) ).

% plus_real_def
tff(fact_9868_cr__real__def,axiom,
    ! [X5: fun_nat_rat,Xa2: real] :
      ( pp(aa_real_bool(aa_fun5258240734392065441l_bool(cr_real,X5),Xa2))
    <=> ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X5),X5))
        & ( aa_fun_nat_rat_real(real2,X5) = Xa2 ) ) ) ).

% cr_real_def
tff(fact_9869_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M7: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_cz(F),M7),at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,M4))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_9870_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_9871_Bseq__mono__convergent,axiom,
    ! [X7: fun_nat_real] :
      ( bfun_nat_real(X7,at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X7,M4)),aa_nat_real(X7,N2))) )
       => topolo7531315842566124627t_real(X7) ) ) ).

% Bseq_mono_convergent
tff(fact_9872_convergent__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))
       => topolo7531315842566124627t_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% convergent_realpow
tff(fact_9873_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M7: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_cz(F),M7),at_top_nat)
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M4),N2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,M4)),aa_nat_real(F,N2))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_9874_at__right__to__0,axiom,
    ! [A: real] : topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)) = filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_vk,A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_right_to_0
tff(fact_9875_at__right__to__top,axiom,
    topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)) = filtermap_real_real(inverse_inverse_real,at_top_real) ).

% at_right_to_top
tff(fact_9876_at__top__to__right,axiom,
    at_top_real = filtermap_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_top_to_right
tff(fact_9877_filtermap__ln__at__right,axiom,
    filtermap_real_real(ln_ln_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) = at_bot_real ).

% filtermap_ln_at_right
tff(fact_9878_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: 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,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_9879_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ).

% pair_less_iff1
tff(fact_9880_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_9881_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_tk),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_9882_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_ii,aTP_Lamm_hw),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_9883_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: 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,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_9884_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9885_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(Y,YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_9886_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(X,XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_9887_wmin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X7,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_9888_wmax__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(X7)
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X7),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_9889_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(Y,Y6)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X7,Y6),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X7),Y6),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_9890_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(X,XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_9891_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(Y6)
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y6),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_9892_smin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( X7 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X7,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_9893_min__weak__def,axiom,
    fun_min_weak = sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_9894_max__weak__def,axiom,
    fun_max_weak = sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_9895_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_9896_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_9897_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9898_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9899_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi2905341329935302413cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_9900_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N),transi6264000038957366511cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% less_eq
tff(fact_9901_euclidean__size__int__def,axiom,
    euclid4774559944035922753ze_int = comp_int_nat_int(nat2,abs_abs_int) ).

% euclidean_size_int_def
tff(fact_9902_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(sgn_sgn_int,K) ) ) ).

% division_segment_eq_sgn
tff(fact_9903_division__segment__int__def,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = one_one_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% division_segment_int_def
tff(fact_9904_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% rat_number_expand(5)
tff(fact_9905_less__rat,axiom,
    ! [B: int,D3: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D3 != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D3)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D3)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3)))) ) ) ) ).

% less_rat
tff(fact_9906_add__rat,axiom,
    ! [B: int,D3: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D3 != zero_zero_int )
       => ( aa_rat_rat(plus_plus_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D3)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D3)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3)) ) ) ) ).

% add_rat
tff(fact_9907_le__rat,axiom,
    ! [B: int,D3: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D3 != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D3)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D3)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3)))) ) ) ) ).

% le_rat
tff(fact_9908_diff__rat,axiom,
    ! [B: int,D3: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D3 != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D3)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D3)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D3)) ) ) ) ).

% diff_rat
tff(fact_9909_eq__rat_I1_J,axiom,
    ! [B: int,D3: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D3 != zero_zero_int )
       => ( ( aa_int_rat(aa_int_fun_int_rat(fract,A),B) = aa_int_rat(aa_int_fun_int_rat(fract,C),D3) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D3) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9910_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% mult_rat_cancel
tff(fact_9911_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(aa_int_fun_int_rat(fract,A),zero_zero_int) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_9912_eq__rat_I3_J,axiom,
    ! [A: int,C: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),A) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),C) ).

% eq_rat(3)
tff(fact_9913_rat__number__collapse_I6_J,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),zero_zero_int) = zero_zero_rat ).

% rat_number_collapse(6)
tff(fact_9914_rat__number__collapse_I1_J,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),K) = zero_zero_rat ).

% rat_number_collapse(1)
tff(fact_9915_Rat__induct__pos,axiom,
    ! [P: fun_rat_bool,Q3: rat] :
      ( ! [A4: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A4),B3))) )
     => pp(aa_rat_bool(P,Q3)) ) ).

% Rat_induct_pos
tff(fact_9916_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9917_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa_num_rat(numeral_numeral_rat,K) = aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% rat_number_expand(3)
tff(fact_9918_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_num_int(numeral_numeral_int,W)),one_one_int) = aa_num_rat(numeral_numeral_rat,W) ).

% rat_number_collapse(3)
tff(fact_9919_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) = aa_nat_rat(semiri681578069525770553at_rat,K) ).

% Fract_of_nat_eq
tff(fact_9920_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Fract_less_zero_iff
tff(fact_9921_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% zero_less_Fract_iff
tff(fact_9922_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa_int_rat(aa_int_fun_int_rat(fract,Xa),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Xa,X))) ).

% Fract.abs_eq
tff(fact_9923_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% positive_rat
tff(fact_9924_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% Fract_less_one_iff
tff(fact_9925_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_9926_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N)),N) = aa_rat_rat(plus_plus_rat(aa_int_rat(aa_int_fun_int_rat(fract,M),N)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9927_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_us,fract) ).

% Fract.transfer
tff(fact_9928_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% Fract_le_zero_iff
tff(fact_9929_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_Fract_iff
tff(fact_9930_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% Fract_le_one_iff
tff(fact_9931_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% one_le_Fract_iff
tff(fact_9932_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) ).

% rat_number_collapse(4)
tff(fact_9933_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_vl(X)) ).

% quotient_of_def
tff(fact_9934_less__eq__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(M),N))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aTP_Lamm_vm(M),fTrue),N)) ) ).

% less_eq_enat_def
tff(fact_9935_normalize__stable,axiom,
    ! [Q3: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P3),Q3))
       => ( normalize(product_Pair_int_int(P3,Q3)) = product_Pair_int_int(P3,Q3) ) ) ) ).

% normalize_stable
tff(fact_9936_Rat__cases,axiom,
    ! [Q3: rat] :
      ~ ! [A4: int,B3: int] :
          ( ( Q3 = aa_int_rat(aa_int_fun_int_rat(fract,A4),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(algebr932160517623751201me_int,A4),B3)) ) ) ).

% Rat_cases
tff(fact_9937_Rat__induct,axiom,
    ! [P: fun_rat_bool,Q3: rat] :
      ( ! [A4: int,B3: 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(algebr932160517623751201me_int,A4),B3))
           => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A4),B3))) ) )
     => pp(aa_rat_bool(P,Q3)) ) ).

% Rat_induct
tff(fact_9938_Rat__cases__nonzero,axiom,
    ! [Q3: rat] :
      ( ! [A4: int,B3: int] :
          ( ( Q3 = aa_int_rat(aa_int_fun_int_rat(fract,A4),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ( ( A4 != zero_zero_int )
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A4),B3)) ) ) )
     => ( Q3 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9939_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X4: product_prod_int_int] :
      ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(X4)),product_snd_int_int(X4)) )
      & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(X4)))
      & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(X4)),product_snd_int_int(X4)))
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Y4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4))) )
         => ( Y4 = X4 ) ) ) ).

% quotient_of_unique
tff(fact_9940_less__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M),N))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aTP_Lamm_vn(N),fFalse),M)) ) ).

% less_enat_def
tff(fact_9941_set__encode__vimage__Suc,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_9942_set__decode__div__2,axiom,
    ! [X: nat] : 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,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,X)) ).

% set_decode_div_2
tff(fact_9943_coprime__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M),N)) ) ).

% coprime_int_iff
tff(fact_9944_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).

% coprime_nat_abs_right_iff
tff(fact_9945_coprime__nat__abs__left__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% coprime_nat_abs_left_iff
tff(fact_9946_coprime__Suc__0__left,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).

% coprime_Suc_0_left
tff(fact_9947_coprime__Suc__0__right,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_9948_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A3)) = vimage_nat_nat(suc,A3) ).

% vimage_Suc_insert_0
tff(fact_9949_coprime__diff__one__right__nat,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(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_9950_coprime__diff__one__left__nat,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(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_9951_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real))
     => ~ ! [M4: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M4)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M4),N2)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9952_transp__realrel,axiom,
    transp_fun_nat_rat(realrel) ).

% transp_realrel
tff(fact_9953_bdd__above__nat,axiom,
    ! [X7: set_nat] :
      ( pp(aa_set_nat_bool(condit2214826472909112428ve_nat,X7))
    <=> pp(aa_set_nat_bool(finite_finite_nat,X7)) ) ).

% bdd_above_nat
tff(fact_9954_num__of__integer_Otransfer,axiom,
    bNF_re6718328864250387230um_num(code_pcr_integer,fequal_num,comp_nat_num_int(num_of_nat,nat2),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_9955_compute__powr__real,axiom,
    ! [B: real,I: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_vo(B),I)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => ( ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I )
           => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))) ) )
              & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I))))) ) ) ) )
          & ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) != I )
           => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_vo(B),I)) ) ) ) ) ) ).

% compute_powr_real
tff(fact_9956_less__eq__integer_Otransfer,axiom,
    bNF_re6983685936609452663r_bool(code_pcr_integer,bNF_re3324582151780938536l_bool(code_pcr_integer,fequal_bool),ord_less_eq_int,ord_le3102999989581377725nteger) ).

% less_eq_integer.transfer
tff(fact_9957_Code__Target__Nat_ONat_Otransfer,axiom,
    bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,nat2,code_Target_Nat) ).

% Code_Target_Nat.Nat.transfer
tff(fact_9958_nat__of__integer_Otransfer,axiom,
    bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,nat2,code_nat_of_integer) ).

% nat_of_integer.transfer
tff(fact_9959_less__integer_Otransfer,axiom,
    bNF_re6983685936609452663r_bool(code_pcr_integer,bNF_re3324582151780938536l_bool(code_pcr_integer,fequal_bool),ord_less_int,ord_le6747313008572928689nteger) ).

% less_integer.transfer
tff(fact_9960_zero__integer_Otransfer,axiom,
    pp(aa_Code_integer_bool(aa_int3865899974016033897r_bool(code_pcr_integer,zero_zero_int),zero_z3403309356797280102nteger)) ).

% zero_integer.transfer
tff(fact_9961_String_Oempty__neq__Literal,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,S: literal] : zero_zero_literal != literal2(B0,B1,B22,B32,B42,B52,B62,S) ).

% String.empty_neq_Literal
tff(fact_9962_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% int_of_integer_integer_of_nat
tff(fact_9963_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,X)) = aa_nat_int(semiri1314217659103216013at_int,X) ).

% integer_of_nat.rep_eq
tff(fact_9964_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_9965_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : aa_nat_Code_integer(code_integer_of_nat,X) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,X)) ).

% integer_of_nat.abs_eq
tff(fact_9966_integer__of__nat_Otransfer,axiom,
    bNF_re4153400068438556298nteger(fequal_nat,code_pcr_integer,semiri1314217659103216013at_int,code_integer_of_nat) ).

% integer_of_nat.transfer
tff(fact_9967_integer__of__nat__numeral,axiom,
    ! [N: num] : aa_nat_Code_integer(code_integer_of_nat,aa_num_nat(numeral_numeral_nat,N)) = numera6620942414471956472nteger(N) ).

% integer_of_nat_numeral
tff(fact_9968_divmod__nat__code,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = produc8678311845419106900er_nat(code_nat_of_integer,code_nat_of_integer,if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(code_integer_of_nat,M)),zero_z3403309356797280102nteger),produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(code_integer_of_nat,N)),zero_z3403309356797280102nteger),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_nat_Code_integer(code_integer_of_nat,M)),code_divmod_abs(aa_nat_Code_integer(code_integer_of_nat,M),aa_nat_Code_integer(code_integer_of_nat,N))))) ).

% divmod_nat_code
tff(fact_9969_integer__of__nat__def,axiom,
    code_integer_of_nat = map_fu6290471996055670595nteger(id_nat,code_integer_of_int,semiri1314217659103216013at_int) ).

% integer_of_nat_def
tff(fact_9970_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_vp(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_vq(M)) ).

% pairs_le_eq_Sigma
tff(fact_9971_numeral__le__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_le_enat_iff
tff(fact_9972_enat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( extended_enat2(Nat) = extended_enat2(Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% enat.inject
tff(fact_9973_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% enat_ord_simps(2)
tff(fact_9974_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(extended_enat2(M)),extended_enat2(N)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% plus_enat_simps(1)
tff(fact_9975_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extended_enat2(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% enat_ord_simps(1)
tff(fact_9976_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),extended_enat2(zero_zero_nat)) = N ).

% idiff_enat_0_right
tff(fact_9977_idiff__enat__0,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat)),N) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_9978_idiff__enat__enat,axiom,
    ! [A: nat,B: nat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extended_enat2(A)),extended_enat2(B)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% idiff_enat_enat
tff(fact_9979_times__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extended_enat2(N)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% times_enat_simps(1)
tff(fact_9980_max__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,extended_enat2(M)),extended_enat2(N)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_enat_simps(1)
tff(fact_9981_min__enat__simps_I1_J,axiom,
    ! [M: nat,N: nat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,extended_enat2(M)),extended_enat2(N)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_enat_simps(1)
tff(fact_9982_numeral__less__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),extended_enat2(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),N)) ) ).

% numeral_less_enat_iff
tff(fact_9983_finite__enat__bounded,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ! [Y3: extended_enat] :
          ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y3),A3))
         => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Y3),extended_enat2(N))) )
     => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3)) ) ).

% finite_enat_bounded
tff(fact_9984_enat__ile,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),extended_enat2(M)))
     => ? [K2: nat] : N = extended_enat2(K2) ) ).

% enat_ile
tff(fact_9985_enat__iless,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),extended_enat2(M)))
     => ? [K2: nat] : N = extended_enat2(K2) ) ).

% enat_iless
tff(fact_9986_less__enatE,axiom,
    ! [N: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),extended_enat2(M)))
     => ~ ! [K2: nat] :
            ( ( N = extended_enat2(K2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),M)) ) ) ).

% less_enatE
tff(fact_9987_numeral__eq__enat,axiom,
    ! [K: num] : numera1916890842035813515d_enat(K) = extended_enat2(aa_num_nat(numeral_numeral_nat,K)) ).

% numeral_eq_enat
tff(fact_9988_one__enat__def,axiom,
    one_on7984719198319812577d_enat = extended_enat2(one_one_nat) ).

% one_enat_def
tff(fact_9989_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = one_on7984719198319812577d_enat )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(1)
tff(fact_9990_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_on7984719198319812577d_enat = extended_enat2(X) )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(2)
tff(fact_9991_of__nat__eq__enat,axiom,
    ! [N: nat] : semiri4216267220026989637d_enat(N) = extended_enat2(N) ).

% of_nat_eq_enat
tff(fact_9992_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_9993_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_9994_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_9995_Suc__ile__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extended_enat2(aa_nat_nat(suc,M))),N))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),N)) ) ).

% Suc_ile_eq
tff(fact_9996_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),extended_enat2(N)))
    <=> ? [Y7: nat,X10: nat] :
          ( ( X = extended_enat2(X10) )
          & ( Y = extended_enat2(Y7) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X10),Y7)),N)) ) ) ).

% iadd_le_enat_iff
tff(fact_9997_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimnum
tff(fact_9998_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extended_enat2(L)) = vEBT_Node(Info,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_vr(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_9999_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( Y != vEBT_Leaf(A4,B3) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( ( Xa = extend5688581933313929465d_enat )
               => ( Y != vEBT_Node(Info2,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_vr(Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ! [L3: nat] :
                    ( ( Xa = extended_enat2(L3) )
                   => ( Y != vEBT_Node(Info2,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_vr(Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_10000_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_vr(Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_10001_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimcomplete
tff(fact_10002_not__enat__eq,axiom,
    ! [X: extended_enat] :
      ( ! [Y5: nat] : X != extended_enat2(Y5)
    <=> ( X = extend5688581933313929465d_enat ) ) ).

% not_enat_eq
tff(fact_10003_not__infinity__eq,axiom,
    ! [X: extended_enat] :
      ( ( X != extend5688581933313929465d_enat )
    <=> ? [I4: nat] : X = extended_enat2(I4) ) ).

% not_infinity_eq
tff(fact_10004_enat__ord__simps_I6_J,axiom,
    ! [Q3: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),Q3)) ).

% enat_ord_simps(6)
tff(fact_10005_enat__ord__simps_I4_J,axiom,
    ! [Q3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Q3),extend5688581933313929465d_enat))
    <=> ( Q3 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
tff(fact_10006_plus__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Q3),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% plus_enat_simps(3)
tff(fact_10007_plus__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(extend5688581933313929465d_enat),Q3) = extend5688581933313929465d_enat ).

% plus_enat_simps(2)
tff(fact_10008_enat__ord__code_I3_J,axiom,
    ! [Q3: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Q3),extend5688581933313929465d_enat)) ).

% enat_ord_code(3)
tff(fact_10009_enat__ord__simps_I5_J,axiom,
    ! [Q3: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),Q3))
    <=> ( Q3 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
tff(fact_10010_idiff__infinity,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extend5688581933313929465d_enat),N) = extend5688581933313929465d_enat ).

% idiff_infinity
tff(fact_10011_times__enat__simps_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% times_enat_simps(2)
tff(fact_10012_max__enat__simps_I5_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,extend5688581933313929465d_enat),Q3) = extend5688581933313929465d_enat ).

% max_enat_simps(5)
tff(fact_10013_max__enat__simps_I4_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q3),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% max_enat_simps(4)
tff(fact_10014_min__enat__simps_I4_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q3),extend5688581933313929465d_enat) = Q3 ).

% min_enat_simps(4)
tff(fact_10015_min__enat__simps_I5_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,extend5688581933313929465d_enat),Q3) = Q3 ).

% min_enat_simps(5)
tff(fact_10016_idiff__self,axiom,
    ! [N: extended_enat] :
      ( ( N != extend5688581933313929465d_enat )
     => ( aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),N) = zero_z5237406670263579293d_enat ) ) ).

% idiff_self
tff(fact_10017_add__diff__cancel__enat,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != extend5688581933313929465d_enat )
     => ( aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),X) = Y ) ) ).

% add_diff_cancel_enat
tff(fact_10018_idiff__infinity__right,axiom,
    ! [A: nat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extended_enat2(A)),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ).

% idiff_infinity_right
tff(fact_10019_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extended_enat2(N)) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extended_enat2(N)) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
tff(fact_10020_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
tff(fact_10021_Sup__enat__def,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ( A3 = bot_bo7653980558646680370d_enat )
       => ( comple4398354569131411667d_enat(A3) = zero_z5237406670263579293d_enat ) )
      & ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
           => ( comple4398354569131411667d_enat(A3) = lattic921264341876707157d_enat(A3) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
           => ( comple4398354569131411667d_enat(A3) = extend5688581933313929465d_enat ) ) ) ) ) ).

% Sup_enat_def
tff(fact_10022_enat__add__left__cancel__le,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)))
    <=> ( ( A = extend5688581933313929465d_enat )
        | pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),C)) ) ) ).

% enat_add_left_cancel_le
tff(fact_10023_enat__ord__simps_I3_J,axiom,
    ! [Q3: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Q3),extend5688581933313929465d_enat)) ).

% enat_ord_simps(3)
tff(fact_10024_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)))
    <=> ( ( A != extend5688581933313929465d_enat )
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),C)) ) ) ).

% enat_add_left_cancel_less
tff(fact_10025_enat__add__left__cancel,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C) )
    <=> ( ( A = extend5688581933313929465d_enat )
        | ( B = C ) ) ) ).

% enat_add_left_cancel
tff(fact_10026_plus__eq__infty__iff__enat,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N) = extend5688581933313929465d_enat )
    <=> ( ( M = extend5688581933313929465d_enat )
        | ( N = extend5688581933313929465d_enat ) ) ) ).

% plus_eq_infty_iff_enat
tff(fact_10027_infinity__ne__i1,axiom,
    extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).

% infinity_ne_i1
tff(fact_10028_top__enat__def,axiom,
    top_to3028658606643905974d_enat = extend5688581933313929465d_enat ).

% top_enat_def
tff(fact_10029_numeral__ne__infinity,axiom,
    ! [K: num] : numera1916890842035813515d_enat(K) != extend5688581933313929465d_enat ).

% numeral_ne_infinity
tff(fact_10030_Inf__enat__def,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ( A3 = bot_bo7653980558646680370d_enat )
       => ( comple2295165028678016749d_enat(A3) = extend5688581933313929465d_enat ) )
      & ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( comple2295165028678016749d_enat(A3) = ord_Le1955565732374568822d_enat(aTP_Lamm_vs(A3)) ) ) ) ).

% Inf_enat_def
tff(fact_10031_infinity__ne__i0,axiom,
    extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).

% infinity_ne_i0
tff(fact_10032_imult__is__infinity,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) = extend5688581933313929465d_enat )
    <=> ( ( ( A = extend5688581933313929465d_enat )
          & ( B != zero_z5237406670263579293d_enat ) )
        | ( ( B = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
tff(fact_10033_enat__ord__code_I4_J,axiom,
    ! [M: nat] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),extend5688581933313929465d_enat)) ).

% enat_ord_code(4)
tff(fact_10034_less__infinityE,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),extend5688581933313929465d_enat))
     => ~ ! [K2: nat] : N != extended_enat2(K2) ) ).

% less_infinityE
tff(fact_10035_infinity__ilessE,axiom,
    ! [M: nat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),extended_enat2(M))) ).

% infinity_ilessE
tff(fact_10036_enat_Odistinct_I2_J,axiom,
    ! [Nat: nat] : extend5688581933313929465d_enat != extended_enat2(Nat) ).

% enat.distinct(2)
tff(fact_10037_enat_Odistinct_I1_J,axiom,
    ! [Nat: nat] : extended_enat2(Nat) != extend5688581933313929465d_enat ).

% enat.distinct(1)
tff(fact_10038_enat_Oexhaust,axiom,
    ! [Y: extended_enat] :
      ( ! [Nat3: nat] : Y != extended_enat2(Nat3)
     => ( Y = extend5688581933313929465d_enat ) ) ).

% enat.exhaust
tff(fact_10039_enat2__cases,axiom,
    ! [Y: extended_enat,Ya: extended_enat] :
      ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
       => ! [Nata: nat] : Ya != extended_enat2(Nata) )
     => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
         => ( Ya != extend5688581933313929465d_enat ) )
       => ( ( ( Y = extend5688581933313929465d_enat )
           => ! [Nat3: nat] : Ya != extended_enat2(Nat3) )
         => ~ ( ( Y = extend5688581933313929465d_enat )
             => ( Ya != extend5688581933313929465d_enat ) ) ) ) ) ).

% enat2_cases
tff(fact_10040_enat3__cases,axiom,
    ! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
      ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
       => ( ? [Nata: nat] : Ya = extended_enat2(Nata)
         => ! [Natb: nat] : Yb != extended_enat2(Natb) ) )
     => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
         => ( ? [Nata: nat] : Ya = extended_enat2(Nata)
           => ( Yb != extend5688581933313929465d_enat ) ) )
       => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
           => ( ( Ya = extend5688581933313929465d_enat )
             => ! [Nata: nat] : Yb != extended_enat2(Nata) ) )
         => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
             => ( ( Ya = extend5688581933313929465d_enat )
               => ( Yb != extend5688581933313929465d_enat ) ) )
           => ( ( ( Y = extend5688581933313929465d_enat )
               => ( ? [Nat3: nat] : Ya = extended_enat2(Nat3)
                 => ! [Nata: nat] : Yb != extended_enat2(Nata) ) )
             => ( ( ( Y = extend5688581933313929465d_enat )
                 => ( ? [Nat3: nat] : Ya = extended_enat2(Nat3)
                   => ( Yb != extend5688581933313929465d_enat ) ) )
               => ( ( ( Y = extend5688581933313929465d_enat )
                   => ( ( Ya = extend5688581933313929465d_enat )
                     => ! [Nat3: nat] : Yb != extended_enat2(Nat3) ) )
                 => ~ ( ( Y = extend5688581933313929465d_enat )
                     => ( ( Ya = extend5688581933313929465d_enat )
                       => ( Yb != extend5688581933313929465d_enat ) ) ) ) ) ) ) ) ) ) ).

% enat3_cases
tff(fact_10041_enat__ex__split,axiom,
    ! [P: fun_Ex8563670519792538868t_bool] :
      ( ? [X_12: extended_enat] : pp(aa_Ext2772979874139974381t_bool(P,X_12))
    <=> ( pp(aa_Ext2772979874139974381t_bool(P,extend5688581933313929465d_enat))
        | ? [X3: nat] : pp(aa_Ext2772979874139974381t_bool(P,extended_enat2(X3))) ) ) ).

% enat_ex_split
tff(fact_10042_infinity__ileE,axiom,
    ! [M: nat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),extended_enat2(M))) ).

% infinity_ileE
tff(fact_10043_enat__ord__code_I5_J,axiom,
    ! [N: nat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),extended_enat2(N))) ).

% enat_ord_code(5)
tff(fact_10044_plus__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vu(N),extend5688581933313929465d_enat),M) ).

% plus_enat_def
tff(fact_10045_imult__infinity__right,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(N),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_10046_imult__infinity,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),N) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_10047_diff__enat__def,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(A),B) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vw(B),extend5688581933313929465d_enat),A) ).

% diff_enat_def
tff(fact_10048_times__enat__def,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vy(N),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(N),zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),M) ).

% times_enat_def
tff(fact_10049_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa) = Y )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(X,Xa))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = vEBT_Leaf(A4,B3) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Leaf(A4,B3),Xa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Xa = extend5688581933313929465d_enat )
                 => ( ( Y = vEBT_Node(Info2,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_vr(Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info2,Deg2,TreeList2,Summary2),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
                 => ! [L3: nat] :
                      ( ( Xa = extended_enat2(L3) )
                     => ( ( Y = vEBT_Node(Info2,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_vr(Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info2,Deg2,TreeList2,Summary2),extended_enat2(L3))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_10050_the__enat_Osimps,axiom,
    ! [N: nat] : aa_Extended_enat_nat(extended_the_enat,extended_enat2(N)) = N ).

% the_enat.simps
tff(fact_10051_eSuc__Max,axiom,
    ! [A3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A3))
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( aa_Ext7969882492699152700d_enat(extended_eSuc,lattic921264341876707157d_enat(A3)) = lattic921264341876707157d_enat(image_80655429650038917d_enat(extended_eSuc,A3)) ) ) ) ).

% eSuc_Max
tff(fact_10052_eSuc__inject,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,M) = aa_Ext7969882492699152700d_enat(extended_eSuc,N) )
    <=> ( M = N ) ) ).

% eSuc_inject
tff(fact_10053_eSuc__infinity,axiom,
    aa_Ext7969882492699152700d_enat(extended_eSuc,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% eSuc_infinity
tff(fact_10054_eSuc__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N)),aa_Ext7969882492699152700d_enat(extended_eSuc,M)))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),M)) ) ).

% eSuc_mono
tff(fact_10055_eSuc__ile__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N)),aa_Ext7969882492699152700d_enat(extended_eSuc,M)))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),M)) ) ).

% eSuc_ile_mono
tff(fact_10056_eSuc__minus__eSuc,axiom,
    ! [N: extended_enat,M: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N)),aa_Ext7969882492699152700d_enat(extended_eSuc,M)) = aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),M) ).

% eSuc_minus_eSuc
tff(fact_10057_iless__eSuc0,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat)))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
tff(fact_10058_eSuc__minus__1,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N)),one_on7984719198319812577d_enat) = N ).

% eSuc_minus_1
tff(fact_10059_eSuc__numeral,axiom,
    ! [K: num] : aa_Ext7969882492699152700d_enat(extended_eSuc,numera1916890842035813515d_enat(K)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,K),one)) ).

% eSuc_numeral
tff(fact_10060_iless__Suc__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),aa_Ext7969882492699152700d_enat(extended_eSuc,N)))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extended_enat2(M)),N)) ) ).

% iless_Suc_eq
tff(fact_10061_ile__eSuc,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),aa_Ext7969882492699152700d_enat(extended_eSuc,N))) ).

% ile_eSuc
tff(fact_10062_ileI1,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M),N))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,M)),N)) ) ).

% ileI1
tff(fact_10063_eSuc__plus__1,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,N) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(N),one_on7984719198319812577d_enat) ).

% eSuc_plus_1
tff(fact_10064_plus__1__eSuc_I1_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),Q3) = aa_Ext7969882492699152700d_enat(extended_eSuc,Q3) ).

% plus_1_eSuc(1)
tff(fact_10065_plus__1__eSuc_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Q3),one_on7984719198319812577d_enat) = aa_Ext7969882492699152700d_enat(extended_eSuc,Q3) ).

% plus_1_eSuc(2)
tff(fact_10066_mono__eSuc,axiom,
    order_4130057895858720880d_enat(extended_eSuc) ).

% mono_eSuc
tff(fact_10067_iadd__Suc,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,M)),N) = aa_Ext7969882492699152700d_enat(extended_eSuc,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N)) ).

% iadd_Suc
tff(fact_10068_iadd__Suc__right,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),aa_Ext7969882492699152700d_enat(extended_eSuc,N)) = aa_Ext7969882492699152700d_enat(extended_eSuc,aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N)) ).

% iadd_Suc_right
tff(fact_10069_eSuc__max,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_Ext7969882492699152700d_enat(extended_eSuc,X)),aa_Ext7969882492699152700d_enat(extended_eSuc,Y)) ).

% eSuc_max
tff(fact_10070_mult__eSuc__right,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),aa_Ext7969882492699152700d_enat(extended_eSuc,N)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N)) ).

% mult_eSuc_right
tff(fact_10071_mult__eSuc,axiom,
    ! [M: extended_enat,N: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,M)),N) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(N),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N)) ).

% mult_eSuc
tff(fact_10072_zero__ne__eSuc,axiom,
    ! [N: extended_enat] : zero_z5237406670263579293d_enat != aa_Ext7969882492699152700d_enat(extended_eSuc,N) ).

% zero_ne_eSuc
tff(fact_10073_i0__iless__eSuc,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(extended_eSuc,N))) ).

% i0_iless_eSuc
tff(fact_10074_one__eSuc,axiom,
    one_on7984719198319812577d_enat = aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat) ).

% one_eSuc
tff(fact_10075_not__eSuc__ilei0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N)),zero_z5237406670263579293d_enat)) ).

% not_eSuc_ilei0
tff(fact_10076_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( extended_enat2(Y) = aa_Ext7969882492699152700d_enat(extended_eSuc,X) )
    <=> ? [N3: nat] :
          ( ( Y = aa_nat_nat(suc,N3) )
          & ( extended_enat2(N3) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_10077_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,X) = extended_enat2(Y) )
    <=> ? [N3: nat] :
          ( ( Y = aa_nat_nat(suc,N3) )
          & ( X = extended_enat2(N3) ) ) ) ).

% eSuc_enat_iff
tff(fact_10078_eSuc__enat,axiom,
    ! [N: nat] : aa_Ext7969882492699152700d_enat(extended_eSuc,extended_enat2(N)) = extended_enat2(aa_nat_nat(suc,N)) ).

% eSuc_enat
tff(fact_10079_eSuc__Sup,axiom,
    ! [A3: set_Extended_enat] :
      ( ( A3 != bot_bo7653980558646680370d_enat )
     => ( aa_Ext7969882492699152700d_enat(extended_eSuc,comple4398354569131411667d_enat(A3)) = comple4398354569131411667d_enat(image_80655429650038917d_enat(extended_eSuc,A3)) ) ) ).

% eSuc_Sup
tff(fact_10080_eSuc__def,axiom,
    ! [I: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,I) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vz,extend5688581933313929465d_enat),I) ).

% eSuc_def
tff(fact_10081_sub_Otransfer,axiom,
    bNF_re4520166952630509772nteger(fequal_num,bNF_re6501075790457514782nteger(fequal_num,code_pcr_integer),aTP_Lamm_ut,code_sub) ).

% sub.transfer
tff(fact_10082_Code__Numeral_Osub__code_I1_J,axiom,
    aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),one) = zero_z3403309356797280102nteger ).

% Code_Numeral.sub_code(1)
tff(fact_10083_sub_Orep__eq,axiom,
    ! [X: num,Xa: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,X),Xa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Xa)) ).

% sub.rep_eq
tff(fact_10084_sub_Oabs__eq,axiom,
    ! [Xa: num,X: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,Xa),X) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,X))) ).

% sub.abs_eq
tff(fact_10085_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = minus_8373710615458151222nteger(code_dup(aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N)),one_one_Code_integer) ).

% Code_Numeral.sub_code(9)
tff(fact_10086_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(code_dup(aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N)),one_one_Code_integer) ).

% Code_Numeral.sub_code(8)
tff(fact_10087_Code__Numeral_Odup__code_I1_J,axiom,
    code_dup(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% Code_Numeral.dup_code(1)
tff(fact_10088_Code__Numeral_Osub__code_I6_J,axiom,
    ! [M: num,N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = code_dup(aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N)) ).

% Code_Numeral.sub_code(6)
tff(fact_10089_Code__Numeral_Osub__code_I4_J,axiom,
    ! [N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),aa_num_num(bit0,N)) = code_Neg(aa_num_num(bitM,N)) ).

% Code_Numeral.sub_code(4)
tff(fact_10090_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),less_than)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_10091_Code__Numeral_Odup__code_I3_J,axiom,
    ! [N: num] : code_dup(code_Neg(N)) = code_Neg(aa_num_num(bit0,N)) ).

% Code_Numeral.dup_code(3)
tff(fact_10092_less__eq__integer__code_I7_J,axiom,
    ! [K: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,code_Neg(K)),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(7)
tff(fact_10093_less__eq__integer__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),code_Neg(L))) ).

% less_eq_integer_code(3)
tff(fact_10094_less__integer__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),code_Neg(L))) ).

% less_integer_code(3)
tff(fact_10095_less__integer__code_I7_J,axiom,
    ! [K: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,code_Neg(K)),zero_z3403309356797280102nteger)) ).

% less_integer_code(7)
tff(fact_10096_less__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,code_Neg(K)),code_Neg(L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,L),K)) ) ).

% less_integer_code(9)
tff(fact_10097_less__eq__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,code_Neg(K)),code_Neg(L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),K)) ) ).

% less_eq_integer_code(9)
tff(fact_10098_Code__Numeral_Osub__code_I5_J,axiom,
    ! [N: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),aa_num_num(bit1,N)) = code_Neg(aa_num_num(bit0,N)) ).

% Code_Numeral.sub_code(5)
tff(fact_10099_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_10100_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_10101_Quotient__real,axiom,
    quotie6870213296501934825t_real(realrel,real2,rep_real,cr_real) ).

% Quotient_real
tff(fact_10102_Code__Numeral_Osub__code_I2_J,axiom,
    ! [M: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),one) = code_Pos(aa_num_num(bitM,M)) ).

% Code_Numeral.sub_code(2)
tff(fact_10103_Code__Numeral_Odup__code_I2_J,axiom,
    ! [N: num] : code_dup(code_Pos(N)) = code_Pos(aa_num_num(bit0,N)) ).

% Code_Numeral.dup_code(2)
tff(fact_10104_less__eq__integer__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,code_Pos(K)),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(4)
tff(fact_10105_less__eq__integer__code_I2_J,axiom,
    ! [L: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),code_Pos(L))) ).

% less_eq_integer_code(2)
tff(fact_10106_less__integer__code_I2_J,axiom,
    ! [L: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),code_Pos(L))) ).

% less_integer_code(2)
tff(fact_10107_less__integer__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,code_Pos(K)),zero_z3403309356797280102nteger)) ).

% less_integer_code(4)
tff(fact_10108_less__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,code_Pos(K)),code_Pos(L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,K),L)) ) ).

% less_integer_code(5)
tff(fact_10109_one__integer__code,axiom,
    one_one_Code_integer = code_Pos(one) ).

% one_integer_code
tff(fact_10110_Pos__fold_I2_J,axiom,
    ! [K: num] : numera6620942414471956472nteger(aa_num_num(bit0,K)) = code_Pos(aa_num_num(bit0,K)) ).

% Pos_fold(2)
tff(fact_10111_Pos__fold_I1_J,axiom,
    numera6620942414471956472nteger(one) = code_Pos(one) ).

% Pos_fold(1)
tff(fact_10112_less__eq__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,code_Pos(K)),code_Pos(L)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,K),L)) ) ).

% less_eq_integer_code(5)
tff(fact_10113_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_10114_Code__Numeral_Osub__code_I3_J,axiom,
    ! [M: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),one) = code_Pos(aa_num_num(bit0,M)) ).

% Code_Numeral.sub_code(3)
tff(fact_10115_cr__int__def,axiom,
    ! [X5: product_prod_nat_nat] : aa_Pro2928840338864870335t_bool(cr_int,X5) = aa_int_fun_int_bool(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X5)) ).

% cr_int_def
tff(fact_10116_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_10117_Quotient__int,axiom,
    quotie1194848508323700631at_int(intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_10118_gcd__nat_Osemilattice__neutr__axioms,axiom,
    pp(aa_nat_bool(semila9081495762789891438tr_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_10119_max__nat_Osemilattice__neutr__axioms,axiom,
    pp(aa_nat_bool(semila9081495762789891438tr_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_10120_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_10121_Restr__natLeq,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,N)),aTP_Lamm_wa(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wb(N))) ).

% Restr_natLeq
tff(fact_10122_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_10123_Restr__natLeq2,axiom,
    ! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_wc(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_wb(N))) ).

% Restr_natLeq2
tff(fact_10124_natLeq__underS__less,axiom,
    ! [N: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,N)) ).

% natLeq_underS_less
tff(fact_10125_gcd__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.monoid_axioms
tff(fact_10126_max__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.monoid_axioms
tff(fact_10127_infinity__enat__def,axiom,
    extend5688581933313929465d_enat = aa_opt7707970668229209096d_enat(extended_Abs_enat,none_nat) ).

% infinity_enat_def
tff(fact_10128_Abs__enat__cases,axiom,
    ! [X: extended_enat] :
      ~ ! [Y3: option_nat] :
          ( ( X = aa_opt7707970668229209096d_enat(extended_Abs_enat,Y3) )
         => ~ member_option_nat(Y3,top_to8920198386146353926on_nat) ) ).

% Abs_enat_cases
tff(fact_10129_Abs__enat__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,X: extended_enat] :
      ( ! [Y3: option_nat] :
          ( member_option_nat(Y3,top_to8920198386146353926on_nat)
         => pp(aa_Ext2772979874139974381t_bool(P,aa_opt7707970668229209096d_enat(extended_Abs_enat,Y3))) )
     => pp(aa_Ext2772979874139974381t_bool(P,X)) ) ).

% Abs_enat_induct
tff(fact_10130_Abs__enat__inject,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( member_option_nat(X,top_to8920198386146353926on_nat)
     => ( member_option_nat(Y,top_to8920198386146353926on_nat)
       => ( ( aa_opt7707970668229209096d_enat(extended_Abs_enat,X) = aa_opt7707970668229209096d_enat(extended_Abs_enat,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_enat_inject
tff(fact_10131_enat__def,axiom,
    ! [N: nat] : extended_enat2(N) = aa_opt7707970668229209096d_enat(extended_Abs_enat,some_nat(N)) ).

% enat_def
tff(fact_10132_Abs__enat__inverse,axiom,
    ! [Y: option_nat] :
      ( member_option_nat(Y,top_to8920198386146353926on_nat)
     => ( aa_Ext8690348996281249226on_nat(extended_Rep_enat,aa_opt7707970668229209096d_enat(extended_Abs_enat,Y)) = Y ) ) ).

% Abs_enat_inverse
tff(fact_10133_Rep__enat__induct,axiom,
    ! [Y: option_nat,P: fun_option_nat_bool] :
      ( member_option_nat(Y,top_to8920198386146353926on_nat)
     => ( ! [X4: extended_enat] : pp(aa_option_nat_bool(P,aa_Ext8690348996281249226on_nat(extended_Rep_enat,X4)))
       => pp(aa_option_nat_bool(P,Y)) ) ) ).

% Rep_enat_induct
tff(fact_10134_Rep__enat__cases,axiom,
    ! [Y: option_nat] :
      ( member_option_nat(Y,top_to8920198386146353926on_nat)
     => ~ ! [X4: extended_enat] : Y != aa_Ext8690348996281249226on_nat(extended_Rep_enat,X4) ) ).

% Rep_enat_cases
tff(fact_10135_Rep__enat,axiom,
    ! [X: extended_enat] : member_option_nat(aa_Ext8690348996281249226on_nat(extended_Rep_enat,X),top_to8920198386146353926on_nat) ).

% Rep_enat
tff(fact_10136_Rep__enat__inject,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( aa_Ext8690348996281249226on_nat(extended_Rep_enat,X) = aa_Ext8690348996281249226on_nat(extended_Rep_enat,Y) )
    <=> ( X = Y ) ) ).

% Rep_enat_inject
tff(fact_10137_Rep__enat__inverse,axiom,
    ! [X: extended_enat] : aa_opt7707970668229209096d_enat(extended_Abs_enat,aa_Ext8690348996281249226on_nat(extended_Rep_enat,X)) = X ).

% Rep_enat_inverse
tff(fact_10138_type__definition__enat,axiom,
    type_d7649664348572268762on_nat(extended_Rep_enat,extended_Abs_enat,top_to8920198386146353926on_nat) ).

% type_definition_enat
tff(fact_10139_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_10140_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_1
tff(fact_10141_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_2
tff(fact_10142_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_10143_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),nat_list_encode(Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_10144_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat))) ).

% wf_less
tff(fact_10145_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_10146_wf__int__ge__less__than2,axiom,
    ! [D3: int] : wf_int(int_ge_less_than2(D3)) ).

% wf_int_ge_less_than2
tff(fact_10147_wf__int__ge__less__than,axiom,
    ! [D3: int] : wf_int(int_ge_less_than(D3)) ).

% wf_int_ge_less_than
tff(fact_10148_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_10149_max__nat_Ocomm__monoid__axioms,axiom,
    pp(aa_nat_bool(comm_monoid_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.comm_monoid_axioms
tff(fact_10150_gcd__nat_Ocomm__monoid__axioms,axiom,
    pp(aa_nat_bool(comm_monoid_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.comm_monoid_axioms
tff(fact_10151_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)))
             => ( ! [X5: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N2) )
                   => pp(aa_nat_bool(P,Y4)) )
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_10152_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( X = aa_nat_nat(suc,N2) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_wd,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_10153_list__decode_Opsimps_I1_J,axiom,
    ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
   => ( nat_list_decode(zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_10154_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_10155_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,N2) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_wd,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
                 => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_10156_times__num__def,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ).

% times_num_def
tff(fact_10157_less__eq__num__def,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ) ).

% less_eq_num_def
tff(fact_10158_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,X)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X))) ).

% nat_of_num.simps(3)
tff(fact_10159_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(nat_of_num,X))) ).

% nat_of_num_pos
tff(fact_10160_nat__of__num__neq__0,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,X) != zero_zero_nat ).

% nat_of_num_neq_0
tff(fact_10161_nat__of__num__add,axiom,
    ! [X: num,Y: num] : aa_num_nat(nat_of_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,Y)) ).

% nat_of_num_add
tff(fact_10162_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral_nat ).

% nat_of_num_numeral
tff(fact_10163_nat__of__num__inverse,axiom,
    ! [X: num] : aa_nat_num(num_of_nat,aa_num_nat(nat_of_num,X)) = X ).

% nat_of_num_inverse
tff(fact_10164_num__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa_num_nat(nat_of_num,X) = aa_num_nat(nat_of_num,Y) ) ) ).

% num_eq_iff
tff(fact_10165_nat__of__num__code_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = one_one_nat ).

% nat_of_num_code(1)
tff(fact_10166_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X)) ).

% nat_of_num.simps(2)
tff(fact_10167_nat__of__num__inc,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(inc,X)) = aa_nat_nat(suc,aa_num_nat(nat_of_num,X)) ).

% nat_of_num_inc
tff(fact_10168_less__num__def,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ) ).

% less_num_def
tff(fact_10169_nat__of__num__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,N)),aa_num_nat(nat_of_num,N)) ).

% nat_of_num_code(2)
tff(fact_10170_nat__of__num__mult,axiom,
    ! [X: num,Y: num] : aa_num_nat(nat_of_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,Y)) ).

% nat_of_num_mult
tff(fact_10171_nat__of__num__sqr,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(sqr,X)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X)) ).

% nat_of_num_sqr
tff(fact_10172_nat__of__num_Osimps_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_of_num.simps(1)
tff(fact_10173_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(nat_of_num,aa_nat_num(num_of_nat,N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_10174_nat__of__num__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,N)),aa_num_nat(nat_of_num,N))) ).

% nat_of_num_code(3)
tff(fact_10175_plus__num__def,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,M)),aa_num_nat(nat_of_num,N))) ).

% plus_num_def
tff(fact_10176_or__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,X),Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [N2: num] :
                ( ( Xa = aa_num_num(bit0,N2) )
               => ( Y != aa_num_num(bit1,N2) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa = aa_num_num(bit1,N2) )
                 => ( Y != aa_num_num(bit1,N2) ) ) )
           => ( ! [M4: num] :
                  ( ( X = aa_num_num(bit0,M4) )
                 => ( ( Xa = one )
                   => ( Y != aa_num_num(bit1,M4) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa_num_num(bit0,N2) )
                       => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit1,N2) )
                         => ( Y != aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit1,M4) )
                       => ( ( Xa = one )
                         => ( Y != aa_num_num(bit1,M4) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa_num_num(bit0,N2) )
                             => ( Y != aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit1,N2) )
                               => ( Y != aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_num.elims
tff(fact_10177_or__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% or_num.simps(2)
tff(fact_10178_or__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M),N)) ).

% or_num.simps(6)
tff(fact_10179_or__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M),N)) ).

% or_num.simps(8)
tff(fact_10180_or__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,one),aa_num_num(bit1,N)) = aa_num_num(bit1,N) ).

% or_num.simps(3)
tff(fact_10181_or__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,aa_num_num(bit1,M)),one) = aa_num_num(bit1,M) ).

% or_num.simps(7)
tff(fact_10182_or__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M),N)) ).

% or_num.simps(9)
tff(fact_10183_or__num_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,one),one) = one ).

% or_num.simps(1)
tff(fact_10184_or__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M),N)) ).

% or_num.simps(5)
tff(fact_10185_or__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% or_num.simps(4)
tff(fact_10186_or__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,X),Xa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa = aa_num_num(bit0,N2) )
                 => ( ( Y = aa_num_num(bit1,N2) )
                   => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( X = one )
               => ! [N2: num] :
                    ( ( Xa = aa_num_num(bit1,N2) )
                   => ( ( Y = aa_num_num(bit1,N2) )
                     => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa_num_num(bit0,M4) )
                   => ( ( Xa = one )
                     => ( ( Y = aa_num_num(bit1,M4) )
                       => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M4),one)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa_num_num(bit0,N2) )
                         => ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa_num_num(bit1,N2) )
                           => ( ( Y = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ( ( Xa = one )
                           => ( ( Y = aa_num_num(bit1,M4) )
                             => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M4),one)) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa_num_num(bit0,N2) )
                               => ( ( Y = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa_num_num(bit1,N2) )
                                 => ( ( Y = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(bit_un6697907153464112080or_num,M4),N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_num.pelims
tff(fact_10187_or__num__dict,axiom,
    bit_un6697907153464112080or_num = bit_un2785000775030745342or_num ).

% or_num_dict
tff(fact_10188_or__num__rel__dict,axiom,
    bit_un4773296044027857193um_rel = bit_un6909899581280750971um_rel ).

% or_num_rel_dict
tff(fact_10189_unit__factor__simps_I1_J,axiom,
    aa_nat_nat(unit_f2748546683901255202or_nat,zero_zero_nat) = zero_zero_nat ).

% unit_factor_simps(1)
tff(fact_10190_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = one_one_nat ) ) ) ).

% unit_factor_nat_def
tff(fact_10191_VEBT__internal_Ogreater_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_hw),X),Y)) ) ).

% VEBT_internal.greater.simps
tff(fact_10192_VEBT__internal_Ogreater_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_hw),X),Xa)) ) ) ).

% VEBT_internal.greater.elims(1)
tff(fact_10193_VEBT__internal_Ogreater_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_hw),X),Xa)) ) ).

% VEBT_internal.greater.elims(2)
tff(fact_10194_VEBT__internal_Ogreater_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,X),Xa))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(aTP_Lamm_hw),X),Xa)) ) ).

% VEBT_internal.greater.elims(3)
tff(fact_10195_VEBT__internal_Oless_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Y)) ) ).

% VEBT_internal.less.simps
tff(fact_10196_VEBT__internal_Oless_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa)) ) ) ).

% VEBT_internal.less.elims(1)
tff(fact_10197_VEBT__internal_Oless_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa)) ) ).

% VEBT_internal.less.elims(2)
tff(fact_10198_VEBT__internal_Oless_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,X),Xa))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_nat),X),Xa)) ) ).

% VEBT_internal.less.elims(3)
tff(fact_10199_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa)) ) ).

% VEBT_internal.lesseq.elims(3)
tff(fact_10200_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa)) ) ).

% VEBT_internal.lesseq.elims(2)
tff(fact_10201_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa)) ) ) ).

% VEBT_internal.lesseq.elims(1)
tff(fact_10202_VEBT__internal_Olesseq_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Y)) ) ).

% VEBT_internal.lesseq.simps
tff(fact_10203_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
tff(fact_10204_lcm__0__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(gcd_lcm_int(M),N) = zero_zero_int )
    <=> ( ( M = zero_zero_int )
        | ( N = zero_zero_int ) ) ) ).

% lcm_0_iff_int
tff(fact_10205_lcm__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(gcd_lcm_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N)) ).

% lcm_int_int_eq
tff(fact_10206_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_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) ) ) ) ).

% lcm_1_iff_nat
tff(fact_10207_lcm__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N) = aa_int_nat(nat2,aa_int_int(gcd_lcm_int(K),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% lcm_nat_abs_left_eq
tff(fact_10208_ATP_Olambda__1,axiom,
    ! [Uu2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_ur,Uu2) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(Uu2)),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(product_snd_int_int(Uu2),product_fst_int_int(Uu2))) ).

% ATP.lambda_1
tff(fact_10209_ATP_Olambda__2,axiom,
    ! [Uu2: fun_nat_rat] : aa_fun765758790203053448at_rat(aTP_Lamm_ve,Uu2) = if_fun_nat_rat(vanishes(Uu2),aTP_Lamm_va,aTP_Lamm_vd(Uu2)) ).

% ATP.lambda_2
tff(fact_10210_ATP_Olambda__3,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_cb,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,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_3
tff(fact_10211_ATP_Olambda__4,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_pc,Uu2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_4
tff(fact_10212_ATP_Olambda__5,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_pt,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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_5
tff(fact_10213_ATP_Olambda__6,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_oq,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_6
tff(fact_10214_ATP_Olambda__7,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ry,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).

% ATP.lambda_7
tff(fact_10215_ATP_Olambda__8,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_sz,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu2)),Uu2) ).

% ATP.lambda_8
tff(fact_10216_ATP_Olambda__9,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_sb,Uu2) = aa_real_real(aa_nat_fun_real_real(root,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_9
tff(fact_10217_ATP_Olambda__10,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_ul,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_10
tff(fact_10218_ATP_Olambda__11,axiom,
    ! [Uu2: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_uu,Uu2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),zero_zero_nat) ).

% ATP.lambda_11
tff(fact_10219_ATP_Olambda__12,axiom,
    ! [Uu2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_uq,Uu2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_snd_int_int(Uu2)))) ) ).

% ATP.lambda_12
tff(fact_10220_ATP_Olambda__13,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_se,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_13
tff(fact_10221_ATP_Olambda__14,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_rf,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_aq,Uu2)) ).

% ATP.lambda_14
tff(fact_10222_ATP_Olambda__15,axiom,
    ! [Uu2: nat] : aa_nat_set_nat(aTP_Lamm_ra,Uu2) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gw,Uu2)) ).

% ATP.lambda_15
tff(fact_10223_ATP_Olambda__16,axiom,
    ! [Uu2: real] : aa_rea5117059003799598681omplex(aTP_Lamm_up,Uu2) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_uo(Uu2)))) ).

% ATP.lambda_16
tff(fact_10224_ATP_Olambda__17,axiom,
    ! [Uu2: real] : aa_rea5428163979925208277l_real(aTP_Lamm_un,Uu2) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_um(Uu2)))) ).

% ATP.lambda_17
tff(fact_10225_ATP_Olambda__18,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_sg,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).

% ATP.lambda_18
tff(fact_10226_ATP_Olambda__19,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_tw,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).

% ATP.lambda_19
tff(fact_10227_ATP_Olambda__20,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_ts,Uu2) = some_num(aa_num_num(bit0,Uu2)) ).

% ATP.lambda_20
tff(fact_10228_ATP_Olambda__21,axiom,
    ! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_pi,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_21
tff(fact_10229_ATP_Olambda__22,axiom,
    ! [Uu2: nat] : aa_nat_Extended_enat(aTP_Lamm_vz,Uu2) = extended_enat2(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_22
tff(fact_10230_ATP_Olambda__23,axiom,
    ! [Uu2: int] : aa_int_nat(aTP_Lamm_qz,Uu2) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu2)) ).

% ATP.lambda_23
tff(fact_10231_ATP_Olambda__24,axiom,
    ! [Uu2: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_uv,Uu2))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu2,N3))) ) ) ) ).

% ATP.lambda_24
tff(fact_10232_ATP_Olambda__25,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_ub,Uu2))
    <=> ? [I4: int,N3: nat] :
          ( ( Uu2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
          & ( N3 != zero_zero_nat ) ) ) ).

% ATP.lambda_25
tff(fact_10233_ATP_Olambda__26,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_uc,Uu2))
    <=> ? [I4: int,J3: int] :
          ( ( Uu2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_26
tff(fact_10234_ATP_Olambda__27,axiom,
    ! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_tu,Uu2) = some_num(one) ).

% ATP.lambda_27
tff(fact_10235_ATP_Olambda__28,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tz(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_tx(Uua),aTP_Lamm_ty(Uua),Uu2) ).

% ATP.lambda_28
tff(fact_10236_ATP_Olambda__29,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_om(Uu2),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_complex) ).

% ATP.lambda_29
tff(fact_10237_ATP_Olambda__30,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aTP_Lamm_op(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,aa_num_num(bit0,one))),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_real) ).

% ATP.lambda_30
tff(fact_10238_ATP_Olambda__31,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_on(Uu2),Uua) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_rat) ).

% ATP.lambda_31
tff(fact_10239_ATP_Olambda__32,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aTP_Lamm_oo(Uu2),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_int) ).

% ATP.lambda_32
tff(fact_10240_ATP_Olambda__33,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pd(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,aa_num_num(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,aa_num_num(bit0,one))))) ).

% ATP.lambda_33
tff(fact_10241_ATP_Olambda__34,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_px,Uu2),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu2))),one_one_int)) ).

% ATP.lambda_34
tff(fact_10242_ATP_Olambda__35,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_py,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,aa_Code_integer_num(code_num_of_integer,Uu2)),aa_Code_integer_num(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,aa_Code_integer_num(code_num_of_integer,Uu2)),aa_Code_integer_num(code_num_of_integer,Uu2))),one)) ).

% ATP.lambda_35
tff(fact_10243_ATP_Olambda__36,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_qa,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,aa_Code_integer_nat(code_nat_of_integer,Uu2)),aa_Code_integer_nat(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,aa_Code_integer_nat(code_nat_of_integer,Uu2)),aa_Code_integer_nat(code_nat_of_integer,Uu2))),one_one_nat)) ).

% ATP.lambda_36
tff(fact_10244_ATP_Olambda__37,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_su,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(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu2)),Uua),aa_int_int(abs_abs_int,Uu2))) ).

% ATP.lambda_37
tff(fact_10245_ATP_Olambda__38,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_us,Uu2),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Uu2,Uua)) ).

% ATP.lambda_38
tff(fact_10246_ATP_Olambda__39,axiom,
    ! [Uu2: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_vy(Uu2),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vx(Uua),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),Uu2) ).

% ATP.lambda_39
tff(fact_10247_ATP_Olambda__40,axiom,
    ! [Uu2: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_vw(Uu2),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vv(Uua),zero_z5237406670263579293d_enat),Uu2) ).

% ATP.lambda_40
tff(fact_10248_ATP_Olambda__41,axiom,
    ! [Uu2: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_vu(Uu2),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_vt(Uua),extend5688581933313929465d_enat),Uu2) ).

% ATP.lambda_41
tff(fact_10249_ATP_Olambda__42,axiom,
    ! [Uu2: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_vm(Uu2),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(aTP_Lamm_ii,Uua),fFalse),Uu2)) ) ).

% ATP.lambda_42
tff(fact_10250_ATP_Olambda__43,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ip(Uu2),Uua) = aa_complex_complex(if_complex(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_complex) ).

% ATP.lambda_43
tff(fact_10251_ATP_Olambda__44,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aTP_Lamm_is(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_real) ).

% ATP.lambda_44
tff(fact_10252_ATP_Olambda__45,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_iq(Uu2),Uua) = aa_rat_rat(if_rat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_rat) ).

% ATP.lambda_45
tff(fact_10253_ATP_Olambda__46,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aTP_Lamm_ir(Uu2),Uua) = aa_int_int(if_int(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))),zero_zero_int) ).

% ATP.lambda_46
tff(fact_10254_ATP_Olambda__47,axiom,
    ! [Uu2: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_vn(Uu2),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),fTrue),Uu2)) ) ).

% ATP.lambda_47
tff(fact_10255_ATP_Olambda__48,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_ua,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_tz(Uua),Uu2) ).

% ATP.lambda_48
tff(fact_10256_ATP_Olambda__49,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_tx(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ts,bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_49
tff(fact_10257_ATP_Olambda__50,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tt(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ts,bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_50
tff(fact_10258_ATP_Olambda__51,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ap,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_51
tff(fact_10259_ATP_Olambda__52,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pb,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,aa_num_num(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,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_52
tff(fact_10260_ATP_Olambda__53,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pf,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,aa_num_num(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,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_53
tff(fact_10261_ATP_Olambda__54,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_oc(Uu2),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_54
tff(fact_10262_ATP_Olambda__55,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_od(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_55
tff(fact_10263_ATP_Olambda__56,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aTP_Lamm_og(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_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_56
tff(fact_10264_ATP_Olambda__57,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_oe(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_57
tff(fact_10265_ATP_Olambda__58,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aTP_Lamm_of(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_58
tff(fact_10266_ATP_Olambda__59,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pq,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),aa_num_real(numeral_numeral_real,aa_num_num(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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_59
tff(fact_10267_ATP_Olambda__60,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pp,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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_60
tff(fact_10268_ATP_Olambda__61,axiom,
    ! [Uu2: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_pu(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_61
tff(fact_10269_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_qj(Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_62
tff(fact_10270_ATP_Olambda__63,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aq,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,aa_num_num(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,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_63
tff(fact_10271_ATP_Olambda__64,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rg,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,aa_num_num(bit0,one))))) ).

% ATP.lambda_64
tff(fact_10272_ATP_Olambda__65,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rz(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_65
tff(fact_10273_ATP_Olambda__66,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_oh(Uu2),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_66
tff(fact_10274_ATP_Olambda__67,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_oi(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_67
tff(fact_10275_ATP_Olambda__68,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aTP_Lamm_ol(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(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_68
tff(fact_10276_ATP_Olambda__69,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_oj(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_69
tff(fact_10277_ATP_Olambda__70,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aTP_Lamm_ok(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_70
tff(fact_10278_ATP_Olambda__71,axiom,
    ! [Uu2: rat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_vl(Uu2),Uua))
    <=> ( ( Uu2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Uua)))
        & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua))) ) ) ).

% ATP.lambda_71
tff(fact_10279_ATP_Olambda__72,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ou(Uu2),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_72
tff(fact_10280_ATP_Olambda__73,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ow,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_73
tff(fact_10281_ATP_Olambda__74,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ov(Uu2),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_74
tff(fact_10282_ATP_Olambda__75,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_or(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu2),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu2),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_75
tff(fact_10283_ATP_Olambda__76,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ot,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu2),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_76
tff(fact_10284_ATP_Olambda__77,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_os(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu2),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_77
tff(fact_10285_ATP_Olambda__78,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tk,Uu2),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu2),Uua))
        & ( Uu2 != Uua ) ) ) ).

% ATP.lambda_78
tff(fact_10286_ATP_Olambda__79,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk,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,aa_num_num(bit0,one))) ).

% ATP.lambda_79
tff(fact_10287_ATP_Olambda__80,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_tq(Uu2),Uua))
    <=> ( pp(aa_set_int_bool(member_int(Uua),Uu2))
        & ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),Uu2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Uua)) ) ) ) ).

% ATP.lambda_80
tff(fact_10288_ATP_Olambda__81,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pa,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_81
tff(fact_10289_ATP_Olambda__82,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ta,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_82
tff(fact_10290_ATP_Olambda__83,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_83
tff(fact_10291_ATP_Olambda__84,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_hb(Uu2),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).

% ATP.lambda_84
tff(fact_10292_ATP_Olambda__85,axiom,
    ! [Uu2: nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ha(Uu2),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) = one_one_real ) ) ).

% ATP.lambda_85
tff(fact_10293_ATP_Olambda__86,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sl,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_86
tff(fact_10294_ATP_Olambda__87,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ss,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_87
tff(fact_10295_ATP_Olambda__88,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_al,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,aa_num_num(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,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_88
tff(fact_10296_ATP_Olambda__89,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ps,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_89
tff(fact_10297_ATP_Olambda__90,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_bv(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_90
tff(fact_10298_ATP_Olambda__91,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_bw(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_91
tff(fact_10299_ATP_Olambda__92,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_by(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_92
tff(fact_10300_ATP_Olambda__93,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_bx(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_93
tff(fact_10301_ATP_Olambda__94,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ia(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,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_94
tff(fact_10302_ATP_Olambda__95,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hx(Uu2),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_95
tff(fact_10303_ATP_Olambda__96,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hz(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,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_96
tff(fact_10304_ATP_Olambda__97,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hy(Uu2),Uua) = aa_int_int(aa_int_fun_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,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_97
tff(fact_10305_ATP_Olambda__98,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gz(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_98
tff(fact_10306_ATP_Olambda__99,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gx(Uu2),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_99
tff(fact_10307_ATP_Olambda__100,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gy(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,aa_nat_nat(suc,Uua))),aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_100
tff(fact_10308_ATP_Olambda__101,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bf(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_101
tff(fact_10309_ATP_Olambda__102,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_bg(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_102
tff(fact_10310_ATP_Olambda__103,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_el(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_103
tff(fact_10311_ATP_Olambda__104,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cr(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_104
tff(fact_10312_ATP_Olambda__105,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hv(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_105
tff(fact_10313_ATP_Olambda__106,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ht(Uu2),Uua) = aa_rat_rat(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_106
tff(fact_10314_ATP_Olambda__107,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hu(Uu2),Uua) = aa_int_int(aa_int_fun_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_107
tff(fact_10315_ATP_Olambda__108,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hs(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_108
tff(fact_10316_ATP_Olambda__109,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hq(Uu2),Uua) = aa_rat_rat(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_109
tff(fact_10317_ATP_Olambda__110,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hr(Uu2),Uua) = aa_int_int(aa_int_fun_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_110
tff(fact_10318_ATP_Olambda__111,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ki(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_111
tff(fact_10319_ATP_Olambda__112,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kg(Uu2),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uua)),aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_112
tff(fact_10320_ATP_Olambda__113,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kh(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_113
tff(fact_10321_ATP_Olambda__114,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_iz(Uu2),Uua) = aa_real_complex(complex2(aa_complex_real(Uu2,Uua)),zero_zero_real) ).

% ATP.lambda_114
tff(fact_10322_ATP_Olambda__115,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_iy(Uu2),Uua) = aa_real_complex(complex2(aa_nat_real(Uu2,Uua)),zero_zero_real) ).

% ATP.lambda_115
tff(fact_10323_ATP_Olambda__116,axiom,
    ! [Uu2: fun_Ex1451620480498224165t_real,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_et(Uu2),Uua))
    <=> ( aa_Ext2541572767299617110t_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_116
tff(fact_10324_ATP_Olambda__117,axiom,
    ! [Uu2: fun_Ex4152069839811072081at_rat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_ex(Uu2),Uua))
    <=> ( aa_Extended_enat_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_117
tff(fact_10325_ATP_Olambda__118,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_es(Uu2),Uua))
    <=> ( aa_complex_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_118
tff(fact_10326_ATP_Olambda__119,axiom,
    ! [Uu2: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ew(Uu2),Uua))
    <=> ( aa_complex_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_119
tff(fact_10327_ATP_Olambda__120,axiom,
    ! [Uu2: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_eq(Uu2),Uua))
    <=> ( aa_real_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_120
tff(fact_10328_ATP_Olambda__121,axiom,
    ! [Uu2: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_eu(Uu2),Uua))
    <=> ( aa_real_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_121
tff(fact_10329_ATP_Olambda__122,axiom,
    ! [Uu2: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ey(Uu2),Uua))
    <=> ( aa_real_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_122
tff(fact_10330_ATP_Olambda__123,axiom,
    ! [Uu2: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_er(Uu2),Uua))
    <=> ( aa_int_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_123
tff(fact_10331_ATP_Olambda__124,axiom,
    ! [Uu2: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ev(Uu2),Uua))
    <=> ( aa_int_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_124
tff(fact_10332_ATP_Olambda__125,axiom,
    ! [Uu2: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ez(Uu2),Uua))
    <=> ( aa_int_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_125
tff(fact_10333_ATP_Olambda__126,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sr(Uu2),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_126
tff(fact_10334_ATP_Olambda__127,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sq(Uu2),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rz(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,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_127
tff(fact_10335_ATP_Olambda__128,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_tr(Uu2),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu2))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_128
tff(fact_10336_ATP_Olambda__129,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_129
tff(fact_10337_ATP_Olambda__130,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oz,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_130
tff(fact_10338_ATP_Olambda__131,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ox,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_131
tff(fact_10339_ATP_Olambda__132,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_ut,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu2)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_132
tff(fact_10340_ATP_Olambda__133,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_am(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_133
tff(fact_10341_ATP_Olambda__134,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_an,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_134
tff(fact_10342_ATP_Olambda__135,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ao(Uu2),Uua) = aa_rat_rat(plus_plus_rat(Uu2),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_135
tff(fact_10343_ATP_Olambda__136,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nt,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_136
tff(fact_10344_ATP_Olambda__137,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,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_137
tff(fact_10345_ATP_Olambda__138,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_vr(Uu2),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% ATP.lambda_138
tff(fact_10346_ATP_Olambda__139,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_pk,Uu2),Uua) = aa_int_int(aa_int_fun_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_139
tff(fact_10347_ATP_Olambda__140,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(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_140
tff(fact_10348_ATP_Olambda__141,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sh,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_141
tff(fact_10349_ATP_Olambda__142,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rb(Uu2),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_142
tff(fact_10350_ATP_Olambda__143,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bl(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_143
tff(fact_10351_ATP_Olambda__144,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_144
tff(fact_10352_ATP_Olambda__145,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_bn(Uu2),Uua) = aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_145
tff(fact_10353_ATP_Olambda__146,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bp,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_146
tff(fact_10354_ATP_Olambda__147,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_bo(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_147
tff(fact_10355_ATP_Olambda__148,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_wd,Uu2),Uua) = cons_nat(Uu2,nat_list_decode(Uua)) ).

% ATP.lambda_148
tff(fact_10356_ATP_Olambda__149,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jh,Uu2),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu2)) ) ).

% ATP.lambda_149
tff(fact_10357_ATP_Olambda__150,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ji(Uu2),Uua))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Uua),Uu2)) ) ).

% ATP.lambda_150
tff(fact_10358_ATP_Olambda__151,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_jj,Uu2),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu2)) ) ).

% ATP.lambda_151
tff(fact_10359_ATP_Olambda__152,axiom,
    ! [Uu2: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_jk,Uu2),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu2)) ) ).

% ATP.lambda_152
tff(fact_10360_ATP_Olambda__153,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ii,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu2)) ) ).

% ATP.lambda_153
tff(fact_10361_ATP_Olambda__154,axiom,
    ! [Uu2: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jl,Uu2),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu2)) ) ).

% ATP.lambda_154
tff(fact_10362_ATP_Olambda__155,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rd,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu2) ).

% ATP.lambda_155
tff(fact_10363_ATP_Olambda__156,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jg,Uu2),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu2)) ) ).

% ATP.lambda_156
tff(fact_10364_ATP_Olambda__157,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_jc(Uu2),Uua))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(Uua),Uu2)) ) ).

% ATP.lambda_157
tff(fact_10365_ATP_Olambda__158,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_jd,Uu2),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu2)) ) ).

% ATP.lambda_158
tff(fact_10366_ATP_Olambda__159,axiom,
    ! [Uu2: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_je,Uu2),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu2)) ) ).

% ATP.lambda_159
tff(fact_10367_ATP_Olambda__160,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_hw,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu2)) ) ).

% ATP.lambda_160
tff(fact_10368_ATP_Olambda__161,axiom,
    ! [Uu2: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jf,Uu2),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu2)) ) ).

% ATP.lambda_161
tff(fact_10369_ATP_Olambda__162,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sa,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ).

% ATP.lambda_162
tff(fact_10370_ATP_Olambda__163,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(minus_minus_nat,Uua),Uu2) ).

% ATP.lambda_163
tff(fact_10371_ATP_Olambda__164,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rh,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ).

% ATP.lambda_164
tff(fact_10372_ATP_Olambda__165,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_vk,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu2) ).

% ATP.lambda_165
tff(fact_10373_ATP_Olambda__166,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uk,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ).

% ATP.lambda_166
tff(fact_10374_ATP_Olambda__167,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_qb(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu2) ).

% ATP.lambda_167
tff(fact_10375_ATP_Olambda__168,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu2) ).

% ATP.lambda_168
tff(fact_10376_ATP_Olambda__169,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gw,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu2)) ) ).

% ATP.lambda_169
tff(fact_10377_ATP_Olambda__170,axiom,
    ! [Uu2: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jx,Uu2),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu2)) ) ).

% ATP.lambda_170
tff(fact_10378_ATP_Olambda__171,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_qq,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu2) ).

% ATP.lambda_171
tff(fact_10379_ATP_Olambda__172,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sf,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ).

% ATP.lambda_172
tff(fact_10380_ATP_Olambda__173,axiom,
    ! [Uu2: set_Extended_enat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_vs(Uu2),Uua))
    <=> pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uua),Uu2)) ) ).

% ATP.lambda_173
tff(fact_10381_ATP_Olambda__174,axiom,
    ! [Uu2: set_option_nat,Uua: option_nat] :
      ( pp(aa_option_nat_bool(aTP_Lamm_a(Uu2),Uua))
    <=> member_option_nat(Uua,Uu2) ) ).

% ATP.lambda_174
tff(fact_10382_ATP_Olambda__175,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu2),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu2)) ) ).

% ATP.lambda_175
tff(fact_10383_ATP_Olambda__176,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu2),Uua))
    <=> member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_176
tff(fact_10384_ATP_Olambda__177,axiom,
    ! [Uu2: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu2),Uua))
    <=> member_set_nat(Uua,Uu2) ) ).

% ATP.lambda_177
tff(fact_10385_ATP_Olambda__178,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu2),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu2)) ) ).

% ATP.lambda_178
tff(fact_10386_ATP_Olambda__179,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu2),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu2)) ) ).

% ATP.lambda_179
tff(fact_10387_ATP_Olambda__180,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_wc(Uu2),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu2) ).

% ATP.lambda_180
tff(fact_10388_ATP_Olambda__181,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nw(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,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_181
tff(fact_10389_ATP_Olambda__182,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nv(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_182
tff(fact_10390_ATP_Olambda__183,axiom,
    ! [Uu2: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_tg(Uu2),Uua))
    <=> pp(aa_real_bool(Uu2,aa_real_real(inverse_inverse_real,Uua))) ) ).

% ATP.lambda_183
tff(fact_10391_ATP_Olambda__184,axiom,
    ! [Uu2: fun_real_real,Uua: nat] : aa_nat_real(aTP_Lamm_tn(Uu2),Uua) = aa_real_real(Uu2,aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_184
tff(fact_10392_ATP_Olambda__185,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_bh(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_185
tff(fact_10393_ATP_Olambda__186,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ui(Uu2),Uua))
    <=> pp(aa_nat_bool(Uu2,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_186
tff(fact_10394_ATP_Olambda__187,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_bi(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_187
tff(fact_10395_ATP_Olambda__188,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_bk(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_188
tff(fact_10396_ATP_Olambda__189,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_bj(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_189
tff(fact_10397_ATP_Olambda__190,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ty(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_190
tff(fact_10398_ATP_Olambda__191,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tv(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_191
tff(fact_10399_ATP_Olambda__192,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_vc,Uu2),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_vb(Uu2),Uua)) ).

% ATP.lambda_192
tff(fact_10400_ATP_Olambda__193,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qy,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qx(Uu2),Uua)) ).

% ATP.lambda_193
tff(fact_10401_ATP_Olambda__194,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qw,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qv(Uu2),Uua)) ).

% ATP.lambda_194
tff(fact_10402_ATP_Olambda__195,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_qu,Uu2),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_qt(Uu2),Uua)) ).

% ATP.lambda_195
tff(fact_10403_ATP_Olambda__196,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_qs,Uu2),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_qr(Uu2),Uua)) ).

% ATP.lambda_196
tff(fact_10404_ATP_Olambda__197,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qp,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qo(Uu2),Uua)) ).

% ATP.lambda_197
tff(fact_10405_ATP_Olambda__198,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_rq(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_rp(Uu2),Uua)) ).

% ATP.lambda_198
tff(fact_10406_ATP_Olambda__199,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pj(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).

% ATP.lambda_199
tff(fact_10407_ATP_Olambda__200,axiom,
    ! [Uu2: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_tb(Uu2),Uua))
    <=> ( aa_real_real(Uu2,Uua) != zero_zero_real ) ) ).

% ATP.lambda_200
tff(fact_10408_ATP_Olambda__201,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_po,Uu2),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_201
tff(fact_10409_ATP_Olambda__202,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_dc(Uu2),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_202
tff(fact_10410_ATP_Olambda__203,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_dd(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_203
tff(fact_10411_ATP_Olambda__204,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_de(Uu2),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_204
tff(fact_10412_ATP_Olambda__205,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_dg(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_205
tff(fact_10413_ATP_Olambda__206,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_df(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_206
tff(fact_10414_ATP_Olambda__207,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_vq(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).

% ATP.lambda_207
tff(fact_10415_ATP_Olambda__208,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sj,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_208
tff(fact_10416_ATP_Olambda__209,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_vx(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uua)) ).

% ATP.lambda_209
tff(fact_10417_ATP_Olambda__210,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_vv(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).

% ATP.lambda_210
tff(fact_10418_ATP_Olambda__211,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_vt(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)) ).

% ATP.lambda_211
tff(fact_10419_ATP_Olambda__212,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ue,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu2),Uua)) ).

% ATP.lambda_212
tff(fact_10420_ATP_Olambda__213,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ud,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu2)) ).

% ATP.lambda_213
tff(fact_10421_ATP_Olambda__214,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu2),Uua)) ).

% ATP.lambda_214
tff(fact_10422_ATP_Olambda__215,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qh,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu2)) ).

% ATP.lambda_215
tff(fact_10423_ATP_Olambda__216,axiom,
    ! [Uu2: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_ep(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu2,Uua)) ).

% ATP.lambda_216
tff(fact_10424_ATP_Olambda__217,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_cw(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu2,Uua)) ).

% ATP.lambda_217
tff(fact_10425_ATP_Olambda__218,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cy(Uu2),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_218
tff(fact_10426_ATP_Olambda__219,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sd(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_219
tff(fact_10427_ATP_Olambda__220,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_vd(Uu2),Uua) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_220
tff(fact_10428_ATP_Olambda__221,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_di(Uu2),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu2,Uua)) ).

% ATP.lambda_221
tff(fact_10429_ATP_Olambda__222,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_complex(aTP_Lamm_as(Uu2),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_222
tff(fact_10430_ATP_Olambda__223,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_at(Uu2),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_223
tff(fact_10431_ATP_Olambda__224,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_rat(aTP_Lamm_au(Uu2),Uua) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_224
tff(fact_10432_ATP_Olambda__225,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_aw(Uu2),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_225
tff(fact_10433_ATP_Olambda__226,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_av(Uu2),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_226
tff(fact_10434_ATP_Olambda__227,axiom,
    ! [Uu2: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_ar(Uu2),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu2,Uua)) ).

% ATP.lambda_227
tff(fact_10435_ATP_Olambda__228,axiom,
    ! [Uu2: fun_Pr672294278902479040t_real,Uua: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aTP_Lamm_lm(Uu2),Uua) = aa_real_real(ln_ln_real,aa_Pro6168199777849092593t_real(Uu2,Uua)) ).

% ATP.lambda_228
tff(fact_10436_ATP_Olambda__229,axiom,
    ! [Uu2: fun_Ex1451620480498224165t_real,Uua: extended_enat] : aa_Ext2541572767299617110t_real(aTP_Lamm_ln(Uu2),Uua) = aa_real_real(ln_ln_real,aa_Ext2541572767299617110t_real(Uu2,Uua)) ).

% ATP.lambda_229
tff(fact_10437_ATP_Olambda__230,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_ll(Uu2),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu2,Uua)) ).

% ATP.lambda_230
tff(fact_10438_ATP_Olambda__231,axiom,
    ! [Uu2: fun_option_nat_real,Uua: option_nat] : aa_option_nat_real(aTP_Lamm_lh(Uu2),Uua) = aa_real_real(ln_ln_real,aa_option_nat_real(Uu2,Uua)) ).

% ATP.lambda_231
tff(fact_10439_ATP_Olambda__232,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_li(Uu2),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_232
tff(fact_10440_ATP_Olambda__233,axiom,
    ! [Uu2: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_lj(Uu2),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu2,Uua)) ).

% ATP.lambda_233
tff(fact_10441_ATP_Olambda__234,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lo(Uu2),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_234
tff(fact_10442_ATP_Olambda__235,axiom,
    ! [Uu2: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_lk(Uu2),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu2,Uua)) ).

% ATP.lambda_235
tff(fact_10443_ATP_Olambda__236,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_uy,Uu2),Uua) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_236
tff(fact_10444_ATP_Olambda__237,axiom,
    ! [Uu2: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_dj(Uu2),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu2,Uua)) ).

% ATP.lambda_237
tff(fact_10445_ATP_Olambda__238,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_complex(aTP_Lamm_ay(Uu2),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_238
tff(fact_10446_ATP_Olambda__239,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_ax(Uu2),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_239
tff(fact_10447_ATP_Olambda__240,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_az(Uu2),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_240
tff(fact_10448_ATP_Olambda__241,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ba(Uu2),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_241
tff(fact_10449_ATP_Olambda__242,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_complex(aTP_Lamm_bc(Uu2),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_242
tff(fact_10450_ATP_Olambda__243,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_bb(Uu2),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_243
tff(fact_10451_ATP_Olambda__244,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_bd(Uu2),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_244
tff(fact_10452_ATP_Olambda__245,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_be(Uu2),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_245
tff(fact_10453_ATP_Olambda__246,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_hf(Uu2),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_246
tff(fact_10454_ATP_Olambda__247,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_hc(Uu2),Uua) = plus_plus_rat(aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_247
tff(fact_10455_ATP_Olambda__248,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_he(Uu2),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_248
tff(fact_10456_ATP_Olambda__249,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_hd(Uu2),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_249
tff(fact_10457_ATP_Olambda__250,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ti(Uu2),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_250
tff(fact_10458_ATP_Olambda__251,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cx(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_251
tff(fact_10459_ATP_Olambda__252,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ee(Uu2),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_252
tff(fact_10460_ATP_Olambda__253,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sp(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_253
tff(fact_10461_ATP_Olambda__254,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_rl(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ).

% ATP.lambda_254
tff(fact_10462_ATP_Olambda__255,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_wa(Uu2),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hw,Uu2)) ).

% ATP.lambda_255
tff(fact_10463_ATP_Olambda__256,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_uj(Uu2),Uua))
    <=> ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),Uu2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Uua)) ) ) ).

% ATP.lambda_256
tff(fact_10464_ATP_Olambda__257,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_vi(Uu2),Uua) = aa_nat_rat(semiri681578069525770553at_rat,Uu2) ).

% ATP.lambda_257
tff(fact_10465_ATP_Olambda__258,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_rat(aTP_Lamm_vh(Uu2),Uua) = aa_int_rat(ring_1_of_int_rat,Uu2) ).

% ATP.lambda_258
tff(fact_10466_ATP_Olambda__259,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pe(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,aa_num_num(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,aa_num_num(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,aa_num_num(bit0,one))))) ).

% ATP.lambda_259
tff(fact_10467_ATP_Olambda__260,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nu(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,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_260
tff(fact_10468_ATP_Olambda__261,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_pw(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu2)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_261
tff(fact_10469_ATP_Olambda__262,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_pg(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,aa_num_num(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,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_262
tff(fact_10470_ATP_Olambda__263,axiom,
    ! [Uu2: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ph(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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu2))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_263
tff(fact_10471_ATP_Olambda__264,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ef(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_264
tff(fact_10472_ATP_Olambda__265,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_eg(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_265
tff(fact_10473_ATP_Olambda__266,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_eh(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_266
tff(fact_10474_ATP_Olambda__267,axiom,
    ! [Uu2: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_dn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uu2),Uub)),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_267
tff(fact_10475_ATP_Olambda__268,axiom,
    ! [Uu2: extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: extended_enat] : aa_Extended_enat_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_ds(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uu2),Uub),aa_Extended_enat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_268
tff(fact_10476_ATP_Olambda__269,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_dm(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(fequal_complex(Uu2),Uub)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_269
tff(fact_10477_ATP_Olambda__270,axiom,
    ! [Uu2: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_dr(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(fequal_complex(Uu2),Uub),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_270
tff(fact_10478_ATP_Olambda__271,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_dk(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_271
tff(fact_10479_ATP_Olambda__272,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_do(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu2),Uub),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_272
tff(fact_10480_ATP_Olambda__273,axiom,
    ! [Uu2: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_dt(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu2),Uub)),aa_real_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_273
tff(fact_10481_ATP_Olambda__274,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dp(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu2),Uub),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_274
tff(fact_10482_ATP_Olambda__275,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_dl(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_275
tff(fact_10483_ATP_Olambda__276,axiom,
    ! [Uu2: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_dq(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu2),Uub),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_276
tff(fact_10484_ATP_Olambda__277,axiom,
    ! [Uu2: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_dx(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uub),Uu2)),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_277
tff(fact_10485_ATP_Olambda__278,axiom,
    ! [Uu2: extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: extended_enat] : aa_Extended_enat_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_ec(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uub),Uu2),aa_Extended_enat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_278
tff(fact_10486_ATP_Olambda__279,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_dw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(fequal_complex(Uub),Uu2)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_279
tff(fact_10487_ATP_Olambda__280,axiom,
    ! [Uu2: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_eb(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(fequal_complex(Uub),Uu2),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_280
tff(fact_10488_ATP_Olambda__281,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_du(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_281
tff(fact_10489_ATP_Olambda__282,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dy(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu2),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_282
tff(fact_10490_ATP_Olambda__283,axiom,
    ! [Uu2: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ed(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu2)),aa_real_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_283
tff(fact_10491_ATP_Olambda__284,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cg(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_284
tff(fact_10492_ATP_Olambda__285,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dz(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu2),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_285
tff(fact_10493_ATP_Olambda__286,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ch(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_286
tff(fact_10494_ATP_Olambda__287,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ci(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_287
tff(fact_10495_ATP_Olambda__288,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_dv(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_288
tff(fact_10496_ATP_Olambda__289,axiom,
    ! [Uu2: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ea(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu2),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_289
tff(fact_10497_ATP_Olambda__290,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_qm(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu2),Uub))) ).

% ATP.lambda_290
tff(fact_10498_ATP_Olambda__291,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_qn(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu2),Uub))) ).

% ATP.lambda_291
tff(fact_10499_ATP_Olambda__292,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ql(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu2,Uub))) ).

% ATP.lambda_292
tff(fact_10500_ATP_Olambda__293,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ei(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_293
tff(fact_10501_ATP_Olambda__294,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ej(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_294
tff(fact_10502_ATP_Olambda__295,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ek(Uu2),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu2,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_295
tff(fact_10503_ATP_Olambda__296,axiom,
    ! [Uu2: fun_Ex1451620480498224165t_real,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1090353380788292083t_real(aTP_Lamm_fh(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(Uua,Uub)),aa_Ext2541572767299617110t_real(Uu2,Uub)),zero_zero_real) ).

% ATP.lambda_296
tff(fact_10504_ATP_Olambda__297,axiom,
    ! [Uu2: fun_Ex4152069839811072081at_rat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Extended_enat_rat(aa_fun6658670135733114143at_rat(aTP_Lamm_fn(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(Uua,Uub),aa_Extended_enat_rat(Uu2,Uub)),zero_zero_rat) ).

% ATP.lambda_297
tff(fact_10505_ATP_Olambda__298,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_ff(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_298
tff(fact_10506_ATP_Olambda__299,axiom,
    ! [Uu2: fun_complex_rat,Uua: fun_complex_bool,Uub: complex] : aa_complex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_fm(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(Uua,Uub),aa_complex_rat(Uu2,Uub)),zero_zero_rat) ).

% ATP.lambda_299
tff(fact_10507_ATP_Olambda__300,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_fb(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_300
tff(fact_10508_ATP_Olambda__301,axiom,
    ! [Uu2: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_fi(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu2,Uub)),zero_zero_rat) ).

% ATP.lambda_301
tff(fact_10509_ATP_Olambda__302,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_real_bool,Uub: real] : aa_real_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_fo(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uua,Uub)),aa_real_nat(Uu2,Uub)),zero_zero_nat) ).

% ATP.lambda_302
tff(fact_10510_ATP_Olambda__303,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_fk(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(Uua,Uub),aa_nat_rat(Uu2,Uub)),zero_zero_rat) ).

% ATP.lambda_303
tff(fact_10511_ATP_Olambda__304,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_fd(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_304
tff(fact_10512_ATP_Olambda__305,axiom,
    ! [Uu2: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_fl(Uu2),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(Uua,Uub),aa_int_rat(Uu2,Uub)),zero_zero_rat) ).

% ATP.lambda_305
tff(fact_10513_ATP_Olambda__306,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_tc(Uu2),Uua),Uub))
    <=> has_fi5821293074295781190e_real(Uu2,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_306
tff(fact_10514_ATP_Olambda__307,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rk(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_307
tff(fact_10515_ATP_Olambda__308,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_qk(Uu2),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu2),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_308
tff(fact_10516_ATP_Olambda__309,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_sy(Uu2),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_sx(Uua),Uub)),quotient_of(Uu2))) ) ).

% ATP.lambda_309
tff(fact_10517_ATP_Olambda__310,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_sw(Uu2),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_sv(Uua),Uub)),quotient_of(Uu2))) ) ).

% ATP.lambda_310
tff(fact_10518_ATP_Olambda__311,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rr(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_311
tff(fact_10519_ATP_Olambda__312,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_rs(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_312
tff(fact_10520_ATP_Olambda__313,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_lp(Uu2),Uua),Uub))
    <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cp(Uu2),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_313
tff(fact_10521_ATP_Olambda__314,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_lq(Uu2),Uua),Uub))
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cn(Uu2),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_314
tff(fact_10522_ATP_Olambda__315,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_em(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_315
tff(fact_10523_ATP_Olambda__316,axiom,
    ! [Uu2: fun_nat_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(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_316
tff(fact_10524_ATP_Olambda__317,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_en(Uu2),Uua),Uub) = aa_rat_rat(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_317
tff(fact_10525_ATP_Olambda__318,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ro(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_318
tff(fact_10526_ATP_Olambda__319,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oy(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_319
tff(fact_10527_ATP_Olambda__320,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ug(Uu2),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_320
tff(fact_10528_ATP_Olambda__321,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_uh(Uu2),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_321
tff(fact_10529_ATP_Olambda__322,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_fu(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_322
tff(fact_10530_ATP_Olambda__323,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_fs(Uu2),Uua),Uub))
    <=> ( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub),Uu2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_323
tff(fact_10531_ATP_Olambda__324,axiom,
    ! [Uu2: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_ft(Uu2),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3941691890525107288d_enat(Uub)),Uua)) ) ) ).

% ATP.lambda_324
tff(fact_10532_ATP_Olambda__325,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_fr(Uu2),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(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_325
tff(fact_10533_ATP_Olambda__326,axiom,
    ! [Uu2: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fv(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_326
tff(fact_10534_ATP_Olambda__327,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fx(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_327
tff(fact_10535_ATP_Olambda__328,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_fw(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_328
tff(fact_10536_ATP_Olambda__329,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_uf(Uu2),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_329
tff(fact_10537_ATP_Olambda__330,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_qf(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_330
tff(fact_10538_ATP_Olambda__331,axiom,
    ! [Uu2: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_rc(Uu2),Uua),Uub))
    <=> ( member_set_nat(Uub,pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu2)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_331
tff(fact_10539_ATP_Olambda__332,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_wb(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_332
tff(fact_10540_ATP_Olambda__333,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_qg(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_333
tff(fact_10541_ATP_Olambda__334,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kn(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_334
tff(fact_10542_ATP_Olambda__335,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_pm(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_335
tff(fact_10543_ATP_Olambda__336,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_it(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_336
tff(fact_10544_ATP_Olambda__337,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_pl(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_337
tff(fact_10545_ATP_Olambda__338,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_iw(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_338
tff(fact_10546_ATP_Olambda__339,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ix(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_339
tff(fact_10547_ATP_Olambda__340,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_iu(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_340
tff(fact_10548_ATP_Olambda__341,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ak(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu2),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_341
tff(fact_10549_ATP_Olambda__342,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aj(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu2),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_342
tff(fact_10550_ATP_Olambda__343,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tl(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu2)) ) ) ).

% ATP.lambda_343
tff(fact_10551_ATP_Olambda__344,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_tj(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uu2)) ) ) ).

% ATP.lambda_344
tff(fact_10552_ATP_Olambda__345,axiom,
    ! [Uu2: set_Extended_enat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_fg(Uu2),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu2))
        & pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_345
tff(fact_10553_ATP_Olambda__346,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_fe(Uu2),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu2))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_346
tff(fact_10554_ATP_Olambda__347,axiom,
    ! [Uu2: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_347
tff(fact_10555_ATP_Olambda__348,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_fj(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_348
tff(fact_10556_ATP_Olambda__349,axiom,
    ! [Uu2: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_fc(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_349
tff(fact_10557_ATP_Olambda__350,axiom,
    ! [Uu2: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aTP_Lamm_gg(Uu2),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu2))
        & ( aa_Ext2541572767299617110t_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_350
tff(fact_10558_ATP_Olambda__351,axiom,
    ! [Uu2: set_Extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun5314330785533781769t_bool(aTP_Lamm_gq(Uu2),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu2))
        & ( aa_Extended_enat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_351
tff(fact_10559_ATP_Olambda__352,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_ge(Uu2),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu2))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_352
tff(fact_10560_ATP_Olambda__353,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2759627087638585025x_bool(aTP_Lamm_go(Uu2),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu2))
        & ( aa_complex_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_353
tff(fact_10561_ATP_Olambda__354,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_fy(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_354
tff(fact_10562_ATP_Olambda__355,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_gi(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_355
tff(fact_10563_ATP_Olambda__356,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ga(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_356
tff(fact_10564_ATP_Olambda__357,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_gk(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_357
tff(fact_10565_ATP_Olambda__358,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_gc(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_358
tff(fact_10566_ATP_Olambda__359,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aTP_Lamm_gm(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_359
tff(fact_10567_ATP_Olambda__360,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_pz(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_360
tff(fact_10568_ATP_Olambda__361,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vp(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu2)) ) ).

% ATP.lambda_361
tff(fact_10569_ATP_Olambda__362,axiom,
    ! [Uu2: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_uo(Uu2),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu2)) ) ).

% ATP.lambda_362
tff(fact_10570_ATP_Olambda__363,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_um(Uu2),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu2)) ) ).

% ATP.lambda_363
tff(fact_10571_ATP_Olambda__364,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_iv(Uu2),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).

% ATP.lambda_364
tff(fact_10572_ATP_Olambda__365,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_pn(Uu2),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_365
tff(fact_10573_ATP_Olambda__366,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_cs(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_366
tff(fact_10574_ATP_Olambda__367,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ct(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_367
tff(fact_10575_ATP_Olambda__368,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_so(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,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_368
tff(fact_10576_ATP_Olambda__369,axiom,
    ! [Uu2: fun_nat_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_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_369
tff(fact_10577_ATP_Olambda__370,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_cp(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_370
tff(fact_10578_ATP_Olambda__371,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cn(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_371
tff(fact_10579_ATP_Olambda__372,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_jv(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_372
tff(fact_10580_ATP_Olambda__373,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(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_373
tff(fact_10581_ATP_Olambda__374,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_jw(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_374
tff(fact_10582_ATP_Olambda__375,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fp(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_375
tff(fact_10583_ATP_Olambda__376,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_th(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_376
tff(fact_10584_ATP_Olambda__377,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_td(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu2,Uub)) ).

% ATP.lambda_377
tff(fact_10585_ATP_Olambda__378,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ux,Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_378
tff(fact_10586_ATP_Olambda__379,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pv(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_379
tff(fact_10587_ATP_Olambda__380,axiom,
    ! [Uu2: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_hg(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu2,Uub)) ).

% ATP.lambda_380
tff(fact_10588_ATP_Olambda__381,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_hj(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu2,Uub)) ).

% ATP.lambda_381
tff(fact_10589_ATP_Olambda__382,axiom,
    ! [Uu2: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hl(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu2,Uub)) ).

% ATP.lambda_382
tff(fact_10590_ATP_Olambda__383,axiom,
    ! [Uu2: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_ho(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu2,Uub)) ).

% ATP.lambda_383
tff(fact_10591_ATP_Olambda__384,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_hh(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu2,Uub)) ).

% ATP.lambda_384
tff(fact_10592_ATP_Olambda__385,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hm(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu2,Uub)) ).

% ATP.lambda_385
tff(fact_10593_ATP_Olambda__386,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_hp(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uu2,Uub)) ).

% ATP.lambda_386
tff(fact_10594_ATP_Olambda__387,axiom,
    ! [Uu2: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_hi(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu2,Uub)) ).

% ATP.lambda_387
tff(fact_10595_ATP_Olambda__388,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_hk(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu2,Uub)) ).

% ATP.lambda_388
tff(fact_10596_ATP_Olambda__389,axiom,
    ! [Uu2: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hn(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu2,Uub)) ).

% ATP.lambda_389
tff(fact_10597_ATP_Olambda__390,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sc(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_390
tff(fact_10598_ATP_Olambda__391,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vg(Uu2),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_391
tff(fact_10599_ATP_Olambda__392,axiom,
    ! [Uu2: fun_option_nat_nat,Uua: fun_option_nat_nat,Uub: option_nat] : aa_option_nat_nat(aa_fun7124324411455820056at_nat(aTP_Lamm_js(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_option_nat_nat(Uua,Uub)),aa_option_nat_nat(Uu2,Uub)) ).

% ATP.lambda_392
tff(fact_10600_ATP_Olambda__393,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_ja(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu2,Uub)) ).

% ATP.lambda_393
tff(fact_10601_ATP_Olambda__394,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_jt(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu2,Uub)) ).

% ATP.lambda_394
tff(fact_10602_ATP_Olambda__395,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jb(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu2,Uub)) ).

% ATP.lambda_395
tff(fact_10603_ATP_Olambda__396,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ju(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu2,Uub)) ).

% ATP.lambda_396
tff(fact_10604_ATP_Olambda__397,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_uw,Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_10605_ATP_Olambda__398,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_rn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_398
tff(fact_10606_ATP_Olambda__399,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fq(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua)) ) ).

% ATP.lambda_399
tff(fact_10607_ATP_Olambda__400,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ck(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_400
tff(fact_10608_ATP_Olambda__401,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cl(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_401
tff(fact_10609_ATP_Olambda__402,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_kj(Uu2),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_402
tff(fact_10610_ATP_Olambda__403,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_km(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_403
tff(fact_10611_ATP_Olambda__404,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_kk(Uu2),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uub)),Uua) ).

% ATP.lambda_404
tff(fact_10612_ATP_Olambda__405,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_kl(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_405
tff(fact_10613_ATP_Olambda__406,axiom,
    ! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ri(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_406
tff(fact_10614_ATP_Olambda__407,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cf(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_407
tff(fact_10615_ATP_Olambda__408,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_cd(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_408
tff(fact_10616_ATP_Olambda__409,axiom,
    ! [Uu2: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_ce(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_409
tff(fact_10617_ATP_Olambda__410,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_st(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ).

% ATP.lambda_410
tff(fact_10618_ATP_Olambda__411,axiom,
    ! [Uu2: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rm(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_411
tff(fact_10619_ATP_Olambda__412,axiom,
    ! [Uu2: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_db(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_412
tff(fact_10620_ATP_Olambda__413,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vj(Uu2),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu2,Uub))),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uua,Uub))) ).

% ATP.lambda_413
tff(fact_10621_ATP_Olambda__414,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ic(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_414
tff(fact_10622_ATP_Olambda__415,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ig(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_415
tff(fact_10623_ATP_Olambda__416,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_id(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu2),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_416
tff(fact_10624_ATP_Olambda__417,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_if(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_417
tff(fact_10625_ATP_Olambda__418,axiom,
    ! [Uu2: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ie(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_418
tff(fact_10626_ATP_Olambda__419,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mt(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_419
tff(fact_10627_ATP_Olambda__420,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mw(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_420
tff(fact_10628_ATP_Olambda__421,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mu(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_421
tff(fact_10629_ATP_Olambda__422,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mv(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_422
tff(fact_10630_ATP_Olambda__423,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_si(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_423
tff(fact_10631_ATP_Olambda__424,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_gs(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_424
tff(fact_10632_ATP_Olambda__425,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gv(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_425
tff(fact_10633_ATP_Olambda__426,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_gt(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_426
tff(fact_10634_ATP_Olambda__427,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_gu(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_427
tff(fact_10635_ATP_Olambda__428,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ih(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_428
tff(fact_10636_ATP_Olambda__429,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tf(Uu2),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu2,Uua))) ) ).

% ATP.lambda_429
tff(fact_10637_ATP_Olambda__430,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_430
tff(fact_10638_ATP_Olambda__431,axiom,
    ! [Uu2: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_jo(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_431
tff(fact_10639_ATP_Olambda__432,axiom,
    ! [Uu2: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_jn(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_432
tff(fact_10640_ATP_Olambda__433,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jr(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_433
tff(fact_10641_ATP_Olambda__434,axiom,
    ! [Uu2: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_jp(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_434
tff(fact_10642_ATP_Olambda__435,axiom,
    ! [Uu2: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_jq(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_435
tff(fact_10643_ATP_Olambda__436,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_bq(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_436
tff(fact_10644_ATP_Olambda__437,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_br(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_437
tff(fact_10645_ATP_Olambda__438,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_bs(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_438
tff(fact_10646_ATP_Olambda__439,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bu(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_439
tff(fact_10647_ATP_Olambda__440,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_bt(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_440
tff(fact_10648_ATP_Olambda__441,axiom,
    ! [Uu2: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_vo(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Uu2),Uua) ).

% ATP.lambda_441
tff(fact_10649_ATP_Olambda__442,axiom,
    ! [Uu2: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_te(Uu2),Uua),Uub))
    <=> pp(aa_real_bool(Uu2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua))) ) ).

% ATP.lambda_442
tff(fact_10650_ATP_Olambda__443,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_da(Uu2),Uua),Uub) = aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_443
tff(fact_10651_ATP_Olambda__444,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_cz(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_444
tff(fact_10652_ATP_Olambda__445,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_to(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_445
tff(fact_10653_ATP_Olambda__446,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_cq(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_446
tff(fact_10654_ATP_Olambda__447,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_co(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_447
tff(fact_10655_ATP_Olambda__448,axiom,
    ! [Uu2: fun_op5246294523634652747omplex,Uua: fun_op5246294523634652747omplex,Uub: option_nat] : aa_option_nat_real(aa_fun240986247698212690t_real(aTP_Lamm_la(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_opt3066276567374697100omplex(Uu2,Uub)),aa_opt3066276567374697100omplex(Uua,Uub))) ).

% ATP.lambda_448
tff(fact_10656_ATP_Olambda__449,axiom,
    ! [Uu2: fun_option_nat_real,Uua: fun_option_nat_real,Uub: option_nat] : aa_option_nat_real(aa_fun2975597672206361808t_real(aTP_Lamm_kw(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_option_nat_real(Uu2,Uub)),aa_option_nat_real(Uua,Uub))) ).

% ATP.lambda_449
tff(fact_10657_ATP_Olambda__450,axiom,
    ! [Uu2: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_lb(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(Uu2,Uub)),aa_real_complex(Uua,Uub))) ).

% ATP.lambda_450
tff(fact_10658_ATP_Olambda__451,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kx(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_451
tff(fact_10659_ATP_Olambda__452,axiom,
    ! [Uu2: fun_set_nat_complex,Uua: fun_set_nat_complex,Uub: set_nat] : aa_set_nat_real(aa_fun5767801665285160478t_real(aTP_Lamm_lc(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_set_nat_complex(Uu2,Uub)),aa_set_nat_complex(Uua,Uub))) ).

% ATP.lambda_452
tff(fact_10660_ATP_Olambda__453,axiom,
    ! [Uu2: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_ky(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(Uu2,Uub)),aa_set_nat_real(Uua,Uub))) ).

% ATP.lambda_453
tff(fact_10661_ATP_Olambda__454,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_lf(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(Uua,Uub))) ).

% ATP.lambda_454
tff(fact_10662_ATP_Olambda__455,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_le(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_455
tff(fact_10663_ATP_Olambda__456,axiom,
    ! [Uu2: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_ld(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_int_complex(Uu2,Uub)),aa_int_complex(Uua,Uub))) ).

% ATP.lambda_456
tff(fact_10664_ATP_Olambda__457,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_kz(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Uu2,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_457
tff(fact_10665_ATP_Olambda__458,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn(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_458
tff(fact_10666_ATP_Olambda__459,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_nl(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2)),zero_zero_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_459
tff(fact_10667_ATP_Olambda__460,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nf(Uu2),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),zero_zero_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_460
tff(fact_10668_ATP_Olambda__461,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nj(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2)),zero_zero_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_461
tff(fact_10669_ATP_Olambda__462,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nh(Uu2),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),zero_zero_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_462
tff(fact_10670_ATP_Olambda__463,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_nn(Uu2),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_complex(Uua,Uuc)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_complex),aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_463
tff(fact_10671_ATP_Olambda__464,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_np(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2)),one_one_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_464
tff(fact_10672_ATP_Olambda__465,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_nq(Uu2),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_465
tff(fact_10673_ATP_Olambda__466,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ns(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2)),one_one_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_466
tff(fact_10674_ATP_Olambda__467,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_nr(Uu2),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_467
tff(fact_10675_ATP_Olambda__468,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_no(Uu2),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_468
tff(fact_10676_ATP_Olambda__469,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_nm(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2)),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_469
tff(fact_10677_ATP_Olambda__470,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ng(Uu2),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_470
tff(fact_10678_ATP_Olambda__471,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_nk(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_471
tff(fact_10679_ATP_Olambda__472,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ni(Uu2),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu2),aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_472
tff(fact_10680_ATP_Olambda__473,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_io(Uu2),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_in(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_473
tff(fact_10681_ATP_Olambda__474,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_im(Uu2),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_il(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_474
tff(fact_10682_ATP_Olambda__475,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ik(Uu2),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ij(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_475
tff(fact_10683_ATP_Olambda__476,axiom,
    ! [Uu2: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_nz(Uu2),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_Code_integer,zero_z3403309356797280102nteger))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_476
tff(fact_10684_ATP_Olambda__477,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ny(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_complex_complex(uminus1482373934393186551omplex,Uub)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_complex),zero_zero_complex))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_477
tff(fact_10685_ATP_Olambda__478,axiom,
    ! [Uu2: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ob(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat)),aa_real_real(uminus_uminus_real,Uub)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2)),one_one_real),zero_zero_real))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_478
tff(fact_10686_ATP_Olambda__479,axiom,
    ! [Uu2: nat,Uua: rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_oa(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_rat_rat(uminus_uminus_rat,Uub)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_rat),zero_zero_rat))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_479
tff(fact_10687_ATP_Olambda__480,axiom,
    ! [Uu2: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_nx(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_int_int(uminus_uminus_int,Uub)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu2),one_one_int),zero_zero_int))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_480
tff(fact_10688_ATP_Olambda__481,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_ms(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_mr(Uu2),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_481
tff(fact_10689_ATP_Olambda__482,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_mq(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_mp(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_482
tff(fact_10690_ATP_Olambda__483,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_mo(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_mn(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_483
tff(fact_10691_ATP_Olambda__484,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_mm(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ml(Uu2),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_484
tff(fact_10692_ATP_Olambda__485,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_mj(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_mi(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_485
tff(fact_10693_ATP_Olambda__486,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rx(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_486
tff(fact_10694_ATP_Olambda__487,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rv(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_487
tff(fact_10695_ATP_Olambda__488,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_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,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_488
tff(fact_10696_ATP_Olambda__489,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rt(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_489
tff(fact_10697_ATP_Olambda__490,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lr(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_490
tff(fact_10698_ATP_Olambda__491,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lx(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_491
tff(fact_10699_ATP_Olambda__492,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lt(Uu2),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_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_492
tff(fact_10700_ATP_Olambda__493,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lv(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_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_493
tff(fact_10701_ATP_Olambda__494,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qo(Uu2),Uua),Uub),Uuc) = 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,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_494
tff(fact_10702_ATP_Olambda__495,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ij(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,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_495
tff(fact_10703_ATP_Olambda__496,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_in(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,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_496
tff(fact_10704_ATP_Olambda__497,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_il(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_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,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_497
tff(fact_10705_ATP_Olambda__498,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nb(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uu2),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_498
tff(fact_10706_ATP_Olambda__499,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ne(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_499
tff(fact_10707_ATP_Olambda__500,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nc(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu2),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_500
tff(fact_10708_ATP_Olambda__501,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nd(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu2),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_501
tff(fact_10709_ATP_Olambda__502,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mx(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_502
tff(fact_10710_ATP_Olambda__503,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_na(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,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,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_503
tff(fact_10711_ATP_Olambda__504,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_my(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_504
tff(fact_10712_ATP_Olambda__505,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lg(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_505
tff(fact_10713_ATP_Olambda__506,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_mz(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_506
tff(fact_10714_ATP_Olambda__507,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_lz(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_507
tff(fact_10715_ATP_Olambda__508,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mc(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_508
tff(fact_10716_ATP_Olambda__509,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ma(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_509
tff(fact_10717_ATP_Olambda__510,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_mb(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_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_510
tff(fact_10718_ATP_Olambda__511,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ls(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_511
tff(fact_10719_ATP_Olambda__512,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ly(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_512
tff(fact_10720_ATP_Olambda__513,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lu(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_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_513
tff(fact_10721_ATP_Olambda__514,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lw(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_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_514
tff(fact_10722_ATP_Olambda__515,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_md(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_515
tff(fact_10723_ATP_Olambda__516,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mg(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_516
tff(fact_10724_ATP_Olambda__517,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_me(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_517
tff(fact_10725_ATP_Olambda__518,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_mf(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_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_518
tff(fact_10726_ATP_Olambda__519,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_sv(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_519
tff(fact_10727_ATP_Olambda__520,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_qt(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_520
tff(fact_10728_ATP_Olambda__521,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_sx(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_521
tff(fact_10729_ATP_Olambda__522,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_qr(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_522
tff(fact_10730_ATP_Olambda__523,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qv(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_523
tff(fact_10731_ATP_Olambda__524,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qx(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_524
tff(fact_10732_ATP_Olambda__525,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_vb(Uu2),Uua),Uub),Uuc))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_525
tff(fact_10733_ATP_Olambda__526,axiom,
    ! [Uu2: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_gh(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_526
tff(fact_10734_ATP_Olambda__527,axiom,
    ! [Uu2: set_Extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: fun_Ex4152069839811072081at_rat,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun5314330785533781769t_bool(aa_fun4769541129632704515t_bool(aTP_Lamm_gr(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu2))
        & ( aa_rat_rat(plus_plus_rat(aa_Extended_enat_rat(Uua,Uuc)),aa_Extended_enat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_527
tff(fact_10735_ATP_Olambda__528,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_gf(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(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_528
tff(fact_10736_ATP_Olambda__529,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2759627087638585025x_bool(aa_fun1496239077807485399x_bool(aTP_Lamm_gp(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu2))
        & ( aa_rat_rat(plus_plus_rat(aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_529
tff(fact_10737_ATP_Olambda__530,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_fz(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_530
tff(fact_10738_ATP_Olambda__531,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_gj(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu2))
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_531
tff(fact_10739_ATP_Olambda__532,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_gb(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_532
tff(fact_10740_ATP_Olambda__533,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_gl(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu2))
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_533
tff(fact_10741_ATP_Olambda__534,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_gd(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_534
tff(fact_10742_ATP_Olambda__535,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_gn(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu2))
        & ( aa_rat_rat(plus_plus_rat(aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_535
tff(fact_10743_ATP_Olambda__536,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_cu(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_536
tff(fact_10744_ATP_Olambda__537,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_cv(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_537
tff(fact_10745_ATP_Olambda__538,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ko(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(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,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_538
tff(fact_10746_ATP_Olambda__539,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ku(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(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,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_539
tff(fact_10747_ATP_Olambda__540,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kq(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(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,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_540
tff(fact_10748_ATP_Olambda__541,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ks(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_541
tff(fact_10749_ATP_Olambda__542,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ml(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_542
tff(fact_10750_ATP_Olambda__543,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_mr(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_543
tff(fact_10751_ATP_Olambda__544,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_mn(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_544
tff(fact_10752_ATP_Olambda__545,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_mi(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_545
tff(fact_10753_ATP_Olambda__546,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_mp(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_546
tff(fact_10754_ATP_Olambda__547,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_kf(Uu2),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_ke(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_547
tff(fact_10755_ATP_Olambda__548,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_kd(Uu2),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_kc(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_548
tff(fact_10756_ATP_Olambda__549,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_kb(Uu2),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_ka(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_549
tff(fact_10757_ATP_Olambda__550,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_jz(Uu2),Uua),Uub),Uuc),Uud) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_jy(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_550
tff(fact_10758_ATP_Olambda__551,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_rw(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,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rv(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_551
tff(fact_10759_ATP_Olambda__552,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_kv(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ku(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud)) ).

% ATP.lambda_552
tff(fact_10760_ATP_Olambda__553,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_kt(Uu2),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ks(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_553
tff(fact_10761_ATP_Olambda__554,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_kr(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kq(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_554
tff(fact_10762_ATP_Olambda__555,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_kp(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ko(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_555
tff(fact_10763_ATP_Olambda__556,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_jy(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_556
tff(fact_10764_ATP_Olambda__557,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_ke(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_557
tff(fact_10765_ATP_Olambda__558,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_ka(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_558
tff(fact_10766_ATP_Olambda__559,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_kc(Uu2),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_559
tff(fact_10767_ATP_Olambda__560,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tp(Uu2),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2)))
        & ! [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,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
           => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_12))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I4)) ) )
        & ( ( Uuc = Uud )
         => ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
            & ! [X3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_560
tff(fact_10768_ATP_Olambda__561,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cm,Uu2),Uua) = Uu2 ).

% ATP.lambda_561
tff(fact_10769_ATP_Olambda__562,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_vf(Uu2),Uua) = Uu2 ).

% ATP.lambda_562
tff(fact_10770_ATP_Olambda__563,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_jm,Uu2) = Uu2 ).

% ATP.lambda_563
tff(fact_10771_ATP_Olambda__564,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_ib,Uu2) = Uu2 ).

% ATP.lambda_564
tff(fact_10772_ATP_Olambda__565,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_cc,Uu2) = Uu2 ).

% ATP.lambda_565
tff(fact_10773_ATP_Olambda__566,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_dh,Uu2) = zero_zero_complex ).

% ATP.lambda_566
tff(fact_10774_ATP_Olambda__567,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_af,Uu2) = zero_zero_real ).

% ATP.lambda_567
tff(fact_10775_ATP_Olambda__568,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_ag,Uu2) = zero_zero_rat ).

% ATP.lambda_568
tff(fact_10776_ATP_Olambda__569,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_bz,Uu2) = zero_zero_real ).

% ATP.lambda_569
tff(fact_10777_ATP_Olambda__570,axiom,
    ! [Uu2: nat] : aa_nat_rat(aTP_Lamm_va,Uu2) = zero_zero_rat ).

% ATP.lambda_570
tff(fact_10778_ATP_Olambda__571,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_ah,Uu2) = zero_zero_nat ).

% ATP.lambda_571
tff(fact_10779_ATP_Olambda__572,axiom,
    ! [Uu2: nat] : aa_nat_int(aTP_Lamm_ca,Uu2) = zero_zero_int ).

% ATP.lambda_572
tff(fact_10780_ATP_Olambda__573,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_ai,Uu2) = zero_zero_int ).

% ATP.lambda_573
tff(fact_10781_ATP_Olambda__574,axiom,
    ! [Uu2: nat] : aa_nat_rat(aTP_Lamm_uz,Uu2) = one_one_rat ).

% ATP.lambda_574
tff(fact_10782_ATP_Olambda__575,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_pr,Uu2))
    <=> $false ) ).

% ATP.lambda_575
tff(fact_10783_ATP_Olambda__576,axiom,
    ! [Uu2: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_qe,Uu2))
    <=> $false ) ).

% ATP.lambda_576
tff(fact_10784_ATP_Olambda__577,axiom,
    ! [Uu2: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_qd,Uu2))
    <=> $true ) ).

% ATP.lambda_577

% Helper facts (70)
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(aa_boo8970175072276269406l_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(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_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] : aa_rat_rat(if_rat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fTrue,X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

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__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(fequal_num,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(fequal_num,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(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_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(fequal_complex(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(fequal_complex(X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fTrue,X),Y) = X ).

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] : 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] : if_VEBT_VEBT(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      | pp(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(X),Y))
      | ( X = Y ) ) ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,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_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_If_2_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fTrue,X,Y) = X ).

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_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_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R3: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R3) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R3)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R3: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R3) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R3),aa_VEBT_VEBT_bool(Q,R3)) ).

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 ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R3: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R3) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R3)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R3: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R3) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R3)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R3: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R3) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R3),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    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,aa_num_num(bit0,one))),deg))) ).

%------------------------------------------------------------------------------