%------------------------------------------------------------------------------
% File     : ITP224_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Member 00088_002463
% 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    : 0065_VEBT_Member_00088_002463 [Des22]

% Status   : ContradictoryAxioms
% Rating   : 0.00 v8.2.0, 0.33 v8.1.0
% Syntax   : Number of formulae    : 12849 (5120 unt;2014 typ;   0 def)
%            Number of atoms       : 23052 (9561 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 13415 (1972   ~; 480   |;1044   &)
%                                         (2722 <=>;7197  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   4 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       :  995 ( 517 fml; 478 var)
%            Number of X terms     :  680 (   0  []; 594 ite;  86 let)
%            Number of types       :  328 ( 327 usr)
%            Number of type conns  : 1907 (1281   >; 626   *;   0   +;   0  <<)
%            Number of predicates  :  121 ( 118 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1581 (1581 usr; 418 con; 0-8 aty)
%            Number of variables   : 23089 (22528   !; 561   ?;23089   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 18:30:15.580
%------------------------------------------------------------------------------
% Could-be-implicit typings (327)
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tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_list_nat > fun_list_nat_o ).

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

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

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

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_real_real ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_int_int ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_real_real ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_int_int ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: list_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: list_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: list_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: list_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: list_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: list_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: list_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_real ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_int ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_real > fun_na5225841814499866095t_real ).

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

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

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: real > fun_fu7631928937237476593t_real ).

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

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: nat > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_nat_real ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: complex > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_real_real ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_int_int ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_int_nat > fun_int_int ).

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

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_nat > fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_int_int > fun_int_real ).

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

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: num > fun_num_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_complex_int > fun_fu4213752561377164075ex_int ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_real ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: num > fun_num_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_real > fun_na7718645245721260691t_real ).

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

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: rat > fun_rat_fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_int_int ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_int > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: rat > fun_fu622952192041312191at_rat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_complex_complex > fun_fu8968174646369908141x_real ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_set_nat_complex > fun_fu1617006900395023197t_real ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: nat > fun_fu960360787657196261at_int ).

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

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_real_o ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_o ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_real_o ).

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_o ).

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

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
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tff(sy_c_ATP_058Lamm__uu____,type,
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tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

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tff(sy_c_ATP_058Lamm__ux____,type,
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tff(sy_c_ATP_058Lamm__uy____,type,
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tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_real ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_real_real ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_int_int ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_fun_nat_rat_o ).

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

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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tff(sy_c_Bit__Operations_Otake__bit__num,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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tff(sy_c_Code__Numeral_Ointeger__of__num,type,
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tff(sy_c_Code__Numeral_Onum__of__integer,type,
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tff(sy_c_Code__Target__Int_Onegative,type,
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tff(sy_c_Code__Target__Int_Opositive,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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tff(sy_c_Complex_OArg,type,
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tff(sy_c_Complex_Ocis,type,
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tff(sy_c_Complex_Ocnj,type,
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tff(sy_c_Complex_Ocomplex_OComplex,type,
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tff(sy_c_Complex_Ocomplex_OIm,type,
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tff(sy_c_Complex_Ocomplex_ORe,type,
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tff(sy_c_Complex_Ocsqrt,type,
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tff(sy_c_Complex_Oimaginary__unit,type,
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tff(sy_c_Complex_Orcis,type,
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tff(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Divides_Oadjust__div,type,
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tff(sy_c_Divides_Oadjust__mod,type,
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tff(sy_c_Divides_Odivmod__nat,type,
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tff(sy_c_Divides_Oeucl__rel__int,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
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tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
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tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
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    semiri1408675320244567234ct_nat: fun_nat_nat ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: fun_rat_rat ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: fun_real_real ).

tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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tff(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Finite__Set_Ocard_001_Eo,type,
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tff(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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tff(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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tff(sy_c_List_Oappend_001t__Int__Oint,type,
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tff(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

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,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: ( fun_complex_complex * nat ) > fun_complex_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( fun_int_int * nat ) > fun_int_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( fun_rat_rat * nat * rat ) > rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: fun_real_real > fun_na6724076748692969250l_real ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun_li1025265955262268170_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

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

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

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: ( num * num ) > int ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

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

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

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

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

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

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

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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_001_Eo_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,
    case_option_int_num: ( int * fun_num_int * option_num ) > int ).

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

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

tff(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
    map_option_num_num: ( fun_num_num * option_num ) > option_num ).

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

tff(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,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__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_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,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: ( set_complex * set_complex ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > fun_set_nat_o ).

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

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le4409593196786536943plex_o: ( fun_complex_o * fun_complex_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_less_fun_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_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,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_o ).

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

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

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

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

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

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_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_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

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_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

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__Product____Type__Ounit_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,
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tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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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_001_Eo_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,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,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,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_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Rat_OAbs__Rat,type,
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tff(sy_c_Rat_OFract,type,
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tff(sy_c_Rat_OFrct,type,
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tff(sy_c_Rat_ORep__Rat,type,
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tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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tff(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
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tff(sy_c_Rat_Onormalize,type,
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tff(sy_c_Rat_Oof__int,type,
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tff(sy_c_Rat_Opcr__rat,type,
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tff(sy_c_Rat_Opositive,type,
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tff(sy_c_Rat_Oquotient__of,type,
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tff(sy_c_Rat_Oratrel,type,
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tff(sy_c_Real_Opcr__real,type,
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tff(sy_c_Real_Opositive,type,
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tff(sy_c_Real_Ovanishes,type,
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tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
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tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
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tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osummable_001t__Int__Oint,type,
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tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
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tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osums_001t__Int__Oint,type,
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tff(sy_c_Series_Osums_001t__Nat__Onat,type,
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tff(sy_c_Series_Osums_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001_Eo,type,
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tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_OCollect_001t__Int__Oint,type,
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tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_o > set_list_nat ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
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tff(sy_c_Set_OCollect_001t__Num__Onum,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat * nat ) > fun_complex_complex ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( fun_nat_fun_int_int * nat * nat ) > fun_int_int ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat * rat ) > rat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( fun_na6724076748692969250l_real * nat ) > fun_na6724076748692969250l_real ).

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_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

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_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o * $o ) > 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__Int__Oint,type,
    topolo4899668324122417113eq_int: fun_nat_int > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: fun_nat_nat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: fun_nat_num > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: fun_nat_rat > $o ).

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: fun_nat_set_nat > $o ).

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

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

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: fun_nat_complex > fun_nat_complex ).

tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: fun_nat_int > fun_nat_int ).

tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: fun_nat_rat > fun_nat_rat ).

tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_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_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > 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: vEBT_VEBT > fun_nat_o ).

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_o ).

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_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_o ).

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_o ).

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_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_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li1025265955262268170_nat_o > fun_list_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( fun_Pr9018117161326028078_int_o * product_prod_int_int ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( fun_Pr2028404973800837934_nat_o * product_prod_nat_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( fun_Pr4799464687642144302_num_o * product_prod_num_num ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr8877776288780692068_nat_o * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Code____Numeral__Ointeger,type,
    aa_o_Code_integer: ( fun_o_Code_integer * $o ) > code_integer ).

tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Nat__Onat,type,
    aa_o_nat: ( fun_o_nat * $o ) > nat ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_M_Eo_J,type,
    aa_o_fun_o_o: ( fun_o_fun_o_o * $o ) > fun_o_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001_Eo,type,
    aa_Code_integer_o: ( fun_Code_integer_o * code_integer ) > $o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    aa_Cod1783974652533396728nteger: ( fun_Co8499238673658658103nteger * code_integer ) > code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    aa_Code_integer_int: ( fun_Code_integer_int * code_integer ) > int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    aa_Code_integer_nat: ( fun_Code_integer_nat * code_integer ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    aa_Code_integer_num: ( fun_Code_integer_num * code_integer ) > num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod3168588076859124107eger_o: ( fun_Co1766037875893669450eger_o * code_integer ) > produc6271795597528267376eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod6977894654539884864nteger: ( fun_Co6597186368970283863nteger * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod326357343816142457eger_o: ( fun_Co2081214437782444048eger_o * code_integer ) > fun_Code_integer_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod4161879620478718930nteger: ( fun_Co1981900195452921617nteger * code_integer ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    aa_Cod1242632294114111009er_int: ( fun_Co1092983284983208416er_int * code_integer ) > fun_Code_integer_int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
    aa_Cod6128727250572145423er_num: ( fun_Co8521658056209057742er_num * code_integer ) > fun_Code_integer_num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    aa_Cod1096751537565085541eger_o: ( fun_Co3544761417380347684eger_o * code_integer ) > fun_Co1766037875893669450eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod8380929501330563836nteger: ( fun_Co2470855966376689427nteger * code_integer ) > fun_Co6597186368970283863nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_J,type,
    aa_Cod6397266357770349983nteger: ( fun_Co8502721031311864414nteger * code_integer ) > fun_Co6226381647958292740nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1578151519121547077nteger: ( fun_Co6226381647958292740nteger * code_integer ) > fun_nat_Code_integer ).

tff(sy_c_aa_001t__Complex__Ocomplex_001_Eo,type,
    aa_complex_o: ( fun_complex_o * complex ) > $o ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
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    aa_fun5140708168904641446at_int: ( fun_fu960360787657196261at_int * fun_nat_int ) > fun_fu8464300013002972271at_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
    aa_fun_nat_nat_o: ( fun_fun_nat_nat_o * fun_nat_nat ) > $o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_fun1958354139826840918omplex: ( fun_fu4781654129426620053omplex * fun_nat_nat ) > fun_nat_complex ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    aa_fun1613598493704395220at_int: ( fun_fu4512855384587262995at_int * fun_nat_nat ) > fun_nat_int ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun5791449513213591928at_nat: ( fun_fu7002938529518989751at_nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    aa_fun2283220954820195200at_rat: ( fun_fu622952192041312191at_rat * fun_nat_nat ) > fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun1538629612289296340t_real: ( fun_fu3106432010075488275t_real * fun_nat_nat ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6650658898577385413at_nat: ( fun_fu4413992040155564420at_nat * fun_nat_nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    aa_fun2341495900663639033_nat_o: ( fun_fu9201817095025044880_nat_o * fun_nat_nat ) > fun_fun_nat_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun6854695580826382482at_nat: ( fun_fu5935951186182374353at_nat * fun_nat_nat ) > fun_fu7002938529518989751at_nat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo,type,
    aa_fun_nat_rat_o: ( fun_fun_nat_rat_o * fun_nat_rat ) > $o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    aa_fun765758790203053448at_rat: ( fun_fu8328862064278946247at_rat * fun_nat_rat ) > fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_fun2663889555218712533at_rat: ( fun_fu6228432884689642900at_rat * fun_nat_rat ) > fun_nat_fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_fun5559679822142647773at_rat: ( fun_fu4710970720072501148at_rat * fun_nat_rat ) > fun_rat_fun_nat_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    aa_fun8528462509625601970at_rat: ( fun_fu5304001865264412401at_rat * fun_nat_rat ) > fun_ra3995598502166916815at_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun8455130352500885026real_o: ( fun_fu1771228864347509049real_o * fun_nat_rat ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    aa_fun3638354034882855594at_rat: ( fun_fu6579618355643354601at_rat * fun_nat_rat ) > fun_fu8328862064278946247at_rat ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun1839206346889086768t_real: ( fun_fu6899955116716589935t_real * fun_nat_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    aa_fun3023522663955195517t_real: ( fun_fu163975356400620604t_real * fun_nat_real ) > fun_nat_fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    aa_fun4732485501138722137t_real: ( fun_fu8064956510386623256t_real * fun_nat_real ) > fun_re4238438383274954274t_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    aa_fun2548873334967081986t_real: ( fun_fu6810213084838398785t_real * fun_nat_real ) > fun_re8542384036533109323t_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__fun_It__Nat__Onat_Mt__Real__Oreal_J_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    aa_fun8674848580393814182t_real: ( fun_fu1620311477532340197t_real * fun_nat_real ) > fun_fu6899955116716589935t_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun5746040925223427547t_real: ( fun_fu3477889739401271834t_real * fun_na5723442673449079972x_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun241848180832411225t_real: ( fun_fu4540635375566031512t_real * fun_nat_fun_int_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun7039663905532832253t_real: ( fun_fu2115079063411676732t_real * fun_nat_fun_nat_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Rat__Orat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun507281639146497029t_real: ( fun_fu4806068833880117316t_real * fun_nat_fun_rat_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun1113820866546176217t_real: ( fun_fu5589980185390252312t_real * fun_na6724076748692969250l_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
    aa_fun2567993205388704171l_real: ( fun_fu3612569900928539882l_real * fun_na6724076748692969250l_real ) > fun_re4112884445705685835l_real ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_Eo,type,
    aa_fun2650603632653508759_int_o: ( fun_fu7503566767892366382_int_o * fun_Pr2478283820680522243nt_int ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    aa_fun7041730090766645614at_rat: ( fun_fu1098312133097393581at_rat * fun_Pr2478283820680522243nt_int ) > fun_rat_rat ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    aa_fun3734617511155415725_int_o: ( fun_fu5323616038969315652_int_o * fun_Pr2478283820680522243nt_int ) > fun_fu7503566767892366382_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Rat__Orat_Mt__Rat__Orat_J_M_Eo_J,type,
    aa_fun382765352565706371_rat_o: ( fun_fu4306744791715064858_rat_o * fun_Pr2478283820680522243nt_int ) > fun_fun_rat_rat_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    aa_fun3538214411680374423_nat_o: ( fun_fu8391177546919232046_nat_o * fun_Pr2118491707810114563at_nat ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun8399641135911395862nt_int: ( fun_fu3064074204415222357nt_int * fun_Pr2118491707810114563at_nat ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun856177901615811547_int_o: ( fun_fu4638078822479986034_int_o * fun_Pr2118491707810114563at_nat ) > fun_fun_int_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    aa_fun6920661532540581549_nat_o: ( fun_fu3373238862511017284_nat_o * fun_Pr2118491707810114563at_nat ) > fun_fu8391177546919232046_nat_o ).

tff(sy_c_aa_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J_001_Eo,type,
    aa_fun_rat_rat_o: ( fun_fun_rat_rat_o * fun_rat_rat ) > $o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun7923231501384339402nteger: ( fun_fu8643405727515436425nteger * fun_re6978948838062957830nteger ) > fun_re6978948838062957830nteger ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun2786211375485122666l_real: ( fun_fu6344110373156359337l_real * fun_real_complex ) > fun_real_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Int__Oint_J_001t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    aa_fun8270430325945277160al_int: ( fun_fu3737244727311182119al_int * fun_real_int ) > fun_real_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
    aa_fun211602392931233840al_nat: ( fun_fu3214020956574444655al_nat * fun_real_nat ) > fun_real_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
    aa_fun2468082234287226944al_rat: ( fun_fu2598743018846156927al_rat * fun_real_rat ) > fun_real_rat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001_Eo,type,
    aa_fun_real_real_o: ( fun_fun_real_real_o * fun_real_real ) > $o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun2895569901277098738real_o: ( fun_fu3022329567491847689real_o * fun_real_real ) > fun_real_o ).

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__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__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_M_Eo_J,type,
    aa_fun1584707981518017559real_o: ( fun_fu6910828598506988974real_o * fun_fu8328862064278946247at_rat ) > fun_fun_real_real_o ).

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

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_o ).

tff(sy_c_member_001_Eo,type,
    member_o: ( $o * set_o ) > $o ).

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: ( code_integer * set_Code_integer ) > $o ).

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

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_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__Nat__Onat,type,
    member_nat: nat > fun_set_nat_o ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $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__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: ( real * set_real ) > $o ).

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: ( vEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_info____,type,
    info: option4927543243414619207at_nat ).

tff(sy_v_m,type,
    m: nat ).

tff(sy_v_n____,type,
    n: nat ).

tff(sy_v_na,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_ta____,type,
    ta: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

% Relevant facts (10816)
tff(fact_0_Node_OIH_I2_J,axiom,
    ~ aa_nat_o(vEBT_invar_vebt(summary),zero_zero_nat) ).

% Node.IH(2)
tff(fact_1__092_060open_062t_A_092_060in_062_Aset_AtreeList_092_060close_062,axiom,
    member_VEBT_VEBT(ta,set_VEBT_VEBT2(treeList)) ).

% \<open>t \<in> set treeList\<close>
tff(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_At_A_092_060in_062_Aset_AtreeList_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [T: vEBT_VEBT] : ~ member_VEBT_VEBT(T,set_VEBT_VEBT2(treeList)) ).

% \<open>\<And>thesis. (\<And>t. t \<in> set treeList \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_3__092_060open_062invar__vebt_At_An_092_060close_062,axiom,
    aa_nat_o(vEBT_invar_vebt(ta),n) ).

% \<open>invar_vebt t n\<close>
tff(fact_4_Node_OIH_I1_J,axiom,
    ! [X3a: vEBT_VEBT] :
      ( member_VEBT_VEBT(X3a,set_VEBT_VEBT2(treeList))
     => ~ aa_nat_o(vEBT_invar_vebt(X3a),zero_zero_nat) ) ).

% Node.IH(1)
tff(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_Ainvar__vebt_At_An_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [N: nat] : ~ aa_nat_o(vEBT_invar_vebt(ta),N) ).

% \<open>\<And>thesis. (\<And>n. invar_vebt t n \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_6_Node_Oprems,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_Node(info,deg,treeList,summary)),zero_zero_nat) ).

% Node.prems
tff(fact_7_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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_8_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_9_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_10_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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_11_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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_12_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_13_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_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_14_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(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_15_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_16_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(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_17_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_18_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_19_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_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_20_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_21_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_22_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = 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_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na) ) ) ).

% set_n_deg_not_0
tff(fact_23_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_24_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_25_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_26_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_27_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_28_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_29_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_30_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_31_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_32_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_33_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_34_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_35_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_36_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_37_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_38_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_39_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_40_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_41_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_42_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_43_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_44_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_45_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_46_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_47_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_48_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_49_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_50_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_51_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_52_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_53_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_54_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_55_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_56_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_57_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_58_le__zero__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_59_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_60_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_61_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_62_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_63_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_64_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_65_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_66_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_67_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_68_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_69_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_70_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_71_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_72_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_73_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_74_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_75_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_76_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_77_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_78_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_79_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_80_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_81_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_82_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_83_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_84_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_85_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_86_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_87_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_88_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_89_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_90_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_91_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_92_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_93_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_94_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_95_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_96_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_97_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_98_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_99_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_100_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_101_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_102_power__one,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Na) = one_one_nat ).

% power_one
tff(fact_103_power__one,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(one_one_int),Na) = one_one_int ).

% power_one
tff(fact_104_power__one,axiom,
    ! [Na: nat] : aa_nat_real(power_power_real(one_one_real),Na) = one_one_real ).

% power_one
tff(fact_105_power__one,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(one_one_complex),Na) = one_one_complex ).

% power_one
tff(fact_106_power__one,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(one_one_rat),Na) = one_one_rat ).

% power_one
tff(fact_107_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_108_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_109_power__one__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),one_one_nat) = A ).

% power_one_right
tff(fact_110_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_111_power__one__right,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),one_one_nat) = A ).

% power_one_right
tff(fact_112_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_113_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_114_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_115_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_116_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_117_one__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% one_le_power
tff(fact_118_one__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% one_le_power
tff(fact_119_one__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ).

% one_le_power
tff(fact_120_one__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(A),Na)) ) ).

% one_le_power
tff(fact_121_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_122_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_123_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_124_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_125_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_126_power__le__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),one_one_rat) ) ) ).

% power_le_one
tff(fact_127_power__le__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat) ) ) ).

% power_le_one
tff(fact_128_power__le__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),one_one_int) ) ) ).

% power_le_one
tff(fact_129_power__le__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),one_one_real) ) ) ).

% power_le_one
tff(fact_130_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_131_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_132_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(power_power_real(zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_133_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_134_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_135_power__increasing,axiom,
    ! [Na: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_increasing
tff(fact_136_power__increasing,axiom,
    ! [Na: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_increasing
tff(fact_137_power__increasing,axiom,
    ! [Na: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N2)) ) ) ).

% power_increasing
tff(fact_138_power__increasing,axiom,
    ! [Na: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(A),N2)) ) ) ).

% power_increasing
tff(fact_139_power__decreasing,axiom,
    ! [Na: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_140_power__decreasing,axiom,
    ! [Na: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_141_power__decreasing,axiom,
    ! [Na: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_142_power__decreasing,axiom,
    ! [Na: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_143_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_144_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_145_one__power2,axiom,
    aa_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_146_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_147_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_148_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( member_complex(A,collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_149_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( member_real(A,collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_150_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> aa_list_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_151_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_152_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_153_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( member_int(A,collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_154_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_155_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_156_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_157_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_158_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_159_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_160_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X2: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(Q,X2) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_161_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X2: list_nat] :
          ( aa_list_nat_o(P,X2)
        <=> aa_list_nat_o(Q,X2) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_162_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
        <=> aa_set_nat_o(Q,X2) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_163_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_164_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_165_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_166_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_167_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_168_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_169_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_170_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_171_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_172_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_173_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_174_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_175_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_176_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_177_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_178_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_179_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.left_commute
tff(fact_180_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_181_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_182_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_183_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% add.commute
tff(fact_184_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_185_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_186_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_187_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_188_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_189_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_190_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_191_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_192_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_193_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_194_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_195_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_196_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_197_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_198_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_199_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = 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),B2) = 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_200_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_201_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_202_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_203_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),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_204_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_205_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_206_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_207_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_208_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_209_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_210_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_211_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_212_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_213_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_214_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_215_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_216_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_217_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_218_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_219_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_220_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_221_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C2: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% le_iff_add
tff(fact_222_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_223_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_224_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_225_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_226_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_227_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_228_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_229_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_230_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_231_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_mono
tff(fact_232_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_233_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% add_mono
tff(fact_234_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_235_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_236_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(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_237_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(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_238_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(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_239_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_240_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(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_241_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(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_242_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(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_243_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_244_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(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_245_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(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_246_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(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_247_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_248_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_249_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_250_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_251_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_252_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_253_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_254_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_255_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_256_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_257_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_258_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_259_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_260_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_261_power__not__zero,axiom,
    ! [A: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_262_power__not__zero,axiom,
    ! [A: int,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Na) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_263_power__not__zero,axiom,
    ! [A: real,Na: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),Na) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_264_power__not__zero,axiom,
    ! [A: complex,Na: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Na) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_265_power__not__zero,axiom,
    ! [A: rat,Na: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Na) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_266_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_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_267_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_268_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(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_269_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(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_270_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_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_271_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_272_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(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_273_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(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_274_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_275_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(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_276_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_277_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_278_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_279_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(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_280_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(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_281_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_282_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_283_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_284_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_285_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_286_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_287_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_288_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_289_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_290_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_291_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_292_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_293_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_294_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_295_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_296_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_297_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_298_zero__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_power
tff(fact_299_zero__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_le_power
tff(fact_300_zero__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_power
tff(fact_301_zero__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Na)) ) ).

% zero_le_power
tff(fact_302_power__mono,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono
tff(fact_303_power__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ).

% power_mono
tff(fact_304_power__mono,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono
tff(fact_305_power__mono,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na)) ) ) ).

% power_mono
tff(fact_306_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_307_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_308_zero__power2,axiom,
    aa_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_309_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_310_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_311_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_312_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_313_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_314_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_315_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_316_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_317_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_318_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))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_319_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))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_320_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))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_321_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_322_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(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_323_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(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_324_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_325_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_326_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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_327_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_328_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_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_329_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(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_330_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_331_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_332_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_333_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_334_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_335_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_336_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,M: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_337_one__plus__numeral,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_338_one__plus__numeral,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Na)) = numeral_numeral_real(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_339_one__plus__numeral,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(Na)) = numeral_numeral_rat(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_340_one__plus__numeral,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_341_one__plus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_342_numeral__plus__one,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),one_one_complex) = numera6690914467698888265omplex(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_343_numeral__plus__one,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Na)),one_one_real) = numeral_numeral_real(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_344_numeral__plus__one,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Na)),one_one_rat) = numeral_numeral_rat(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_345_numeral__plus__one,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) = aa_num_nat(numeral_numeral_nat,plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_346_numeral__plus__one,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) = aa_num_int(numeral_numeral_int,plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_347_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Na)),one_one_rat)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_348_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_349_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),one_one_int)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_350_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Na)),one_one_real)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_351_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_352_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_real = numeral_numeral_real(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_353_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_rat = numeral_numeral_rat(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_354_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_355_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_356_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera6690914467698888265omplex(Na) = one_one_complex )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_357_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numeral_numeral_real(Na) = one_one_real )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_358_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numeral_numeral_rat(Na) = one_one_rat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_359_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Na) = one_one_nat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_360_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_int(numeral_numeral_int,Na) = one_one_int )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_361_nat__induct2,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,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_o(P,Na) ) ) ) ).

% nat_induct2
tff(fact_362_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: 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),Na)) != 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))),Na) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_363_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: 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),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_364_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: 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),Na)) != 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_365_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: 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),Na)) != 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_366_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_367_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_368_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_369_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_370_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_371_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_372_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_373_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_374_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_375_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_376_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_377_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_378_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(Na)) = numeral_numeral_real(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_379_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Na)) = numeral_numeral_rat(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_380_numeral__plus__numeral,axiom,
    ! [M: num,Na: 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,Na)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_381_numeral__plus__numeral,axiom,
    ! [M: num,Na: 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,Na)) = aa_num_int(numeral_numeral_int,plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_382_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(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_383_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_384_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_385_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,plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_386_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,plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_387_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_388_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_389_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_390_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_391_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_392_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_393_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% is_num_normalize(1)
tff(fact_394_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_395_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa_num_o(ord_less_eq_num(X),one)
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_396_add__One__commute,axiom,
    ! [Na: num] : plus_plus_num(one,Na) = plus_plus_num(Na,one) ).

% add_One_commute
tff(fact_397_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_398_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_399_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% eq_imp_le
tff(fact_400_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_401_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% nat_le_linear
tff(fact_402_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y3: nat] :
                ( aa_nat_o(P,Y3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_403_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_404_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( size_size_list_o(X) != size_size_list_o(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_405_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_406_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_407_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_408_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_409_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_410_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_411_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_412_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_complex != numera6690914467698888265omplex(Na) ).

% zero_neq_numeral
tff(fact_413_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_real != numeral_numeral_real(Na) ).

% zero_neq_numeral
tff(fact_414_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_rat != numeral_numeral_rat(Na) ).

% zero_neq_numeral
tff(fact_415_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Na) ).

% zero_neq_numeral
tff(fact_416_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Na) ).

% zero_neq_numeral
tff(fact_417_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_418_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_419_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_420_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_421_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_422_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_423_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_424_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_425_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ).

% plus_nat.add_0
tff(fact_426_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = M )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_427_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_428_le__add1,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) ).

% le_add1
tff(fact_429_le__add2,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% le_add2
tff(fact_430_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% add_leD1
tff(fact_431_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_432_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_433_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(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_434_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_435_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_436_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_437_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ? [K2: nat] : Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).

% nat_le_iff_add
tff(fact_438_zero__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(Na)) ).

% zero_le_numeral
tff(fact_439_zero__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_le_numeral
tff(fact_440_zero__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_le_numeral
tff(fact_441_zero__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(Na)) ).

% zero_le_numeral
tff(fact_442_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Na)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_443_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_444_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_445_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Na)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_446_one__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(Na)) ).

% one_le_numeral
tff(fact_447_one__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% one_le_numeral
tff(fact_448_one__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) ).

% one_le_numeral
tff(fact_449_one__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(Na)) ).

% one_le_numeral
tff(fact_450_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_451_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_452_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_453_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_454_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_455_numeral__Bit0,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit0,Na)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),numera6690914467698888265omplex(Na)) ).

% numeral_Bit0
tff(fact_456_numeral__Bit0,axiom,
    ! [Na: num] : numeral_numeral_real(aa_num_num(bit0,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Na)),numeral_numeral_real(Na)) ).

% numeral_Bit0
tff(fact_457_numeral__Bit0,axiom,
    ! [Na: num] : numeral_numeral_rat(aa_num_num(bit0,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Na)),numeral_numeral_rat(Na)) ).

% numeral_Bit0
tff(fact_458_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_Bit0
tff(fact_459_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na)) ).

% numeral_Bit0
tff(fact_460_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_461_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_462_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_463_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_464_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_465_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_466_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_467_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_468_semiring__norm_I2_J,axiom,
    plus_plus_num(one,one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_469_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa_num_o(ord_less_eq_num(one),Na) ).

% semiring_norm(68)
tff(fact_470_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(71)
tff(fact_471_semiring__norm_I6_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(aa_num_num(bit0,M),aa_num_num(bit0,Na)) = aa_num_num(bit0,plus_plus_num(M,Na)) ).

% semiring_norm(6)
tff(fact_472_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(1)
tff(fact_473_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_474_semiring__norm_I83_J,axiom,
    ! [Na: num] : one != aa_num_num(bit0,Na) ).

% semiring_norm(83)
tff(fact_475_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_476_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_477_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_478_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_479__092_060open_0620_A_060_Alength_AtreeList_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(treeList)) ).

% \<open>0 < length treeList\<close>
tff(fact_480_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,M: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = 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,Na) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_481_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_482_deg__not__0,axiom,
    ! [Tb: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tb),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% deg_not_0
tff(fact_483_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_484_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_485_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_486_not__gr__zero,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_487_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% numeral_less_iff
tff(fact_488_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% numeral_less_iff
tff(fact_489_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% numeral_less_iff
tff(fact_490_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% numeral_less_iff
tff(fact_491_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_492_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_493_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_494_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_495_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_496_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_497_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_498_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_499_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_500_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_501_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_502_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_503_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_504_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_505_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_eq
tff(fact_506_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_507_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_508_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_509_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_510_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_511_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% Suc_le_mono
tff(fact_512_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc_right
tff(fact_513_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_514_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_515_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_516_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_517_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_518_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_519_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_520_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_521_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_522_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_523_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_524_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_525_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_526_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_527_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_528_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_529_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_530_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_531_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_532_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_533_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_534_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_535_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_536_power__inject__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(power_power_real(A),M) = aa_nat_real(power_power_real(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_537_power__inject__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_538_power__inject__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_539_power__inject__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_540_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Na)) = zero_zero_nat ).

% power_0_Suc
tff(fact_541_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Na)) = zero_zero_int ).

% power_0_Suc
tff(fact_542_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,Na)) = zero_zero_real ).

% power_0_Suc
tff(fact_543_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Na)) = zero_zero_complex ).

% power_0_Suc
tff(fact_544_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Na)) = zero_zero_rat ).

% power_0_Suc
tff(fact_545_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_546_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_547_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_548_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_549_power__Suc0__right,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_550_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_551_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_552_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% add_gr_0
tff(fact_553_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_554_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_555_power__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_556_nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_557_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),numeral_numeral_real(Na))
    <=> aa_num_o(ord_less_num(one),Na) ) ).

% one_less_numeral_iff
tff(fact_558_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(Na))
    <=> aa_num_o(ord_less_num(one),Na) ) ).

% one_less_numeral_iff
tff(fact_559_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(ord_less_num(one),Na) ) ).

% one_less_numeral_iff
tff(fact_560_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(ord_less_num(one),Na) ) ).

% one_less_numeral_iff
tff(fact_561_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),X)),aa_nat_real(power_power_real(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_562_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_563_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_564_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_565_power__eq__0__iff,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_566_power__eq__0__iff,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_nat_int(power_power_int(A),Na) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_567_power__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(power_power_real(A),Na) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_568_power__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Na) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_569_power__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_570_Suc__numeral,axiom,
    ! [Na: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,plus_plus_num(Na,one)) ).

% Suc_numeral
tff(fact_571_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_572_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_573_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_574_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_575_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_576_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_577_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_578_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),X)),aa_nat_real(power_power_real(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_579_power__mono__iff,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_580_power__mono__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_581_power__mono__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_582_power__mono__iff,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_583_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc'
tff(fact_584_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc
tff(fact_585_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_586_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_587_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_588_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_589_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_590_power__decreasing__iff,axiom,
    ! [B: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_591_power__decreasing__iff,axiom,
    ! [B: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_592_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_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_593_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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_594_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_595_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_596_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_597_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_598_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_599_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_600_i0__lb,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Na) ).

% i0_lb
tff(fact_601_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Na),zero_z5237406670263579293d_enat)
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_602_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_603_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_604_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_605_infinite__descent,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Na) ) ).

% infinite_descent
tff(fact_606_nat__less__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% nat_less_induct
tff(fact_607_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_608_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K3)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K3)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K3) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_609_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_610_less__not__refl3,axiom,
    ! [S: nat,Tb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Tb)
     => ( S != Tb ) ) ).

% less_not_refl3
tff(fact_611_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_612_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_613_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_SucD
tff(fact_614_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_615_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M)
    <=> ? [M3: nat] :
          ( ( M = aa_nat_nat(suc,M3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M3) ) ) ).

% Suc_less_eq2
tff(fact_616_All__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,Na)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
           => aa_nat_o(P,I3) ) ) ) ).

% All_less_Suc
tff(fact_617_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_618_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% nat_neq_iff
tff(fact_619_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa_nat_nat(suc,Na) ).

% n_not_Suc_n
tff(fact_620_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_621_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,Na)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
            & aa_nat_o(P,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_622_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_623_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_624_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_625_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( ( aa_nat_nat(suc,M) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_626_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_627_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_lessD
tff(fact_628_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_629_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Na: nat,M: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_630_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Na: nat,M: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_631_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Na: nat,M: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(ord_less_num(aa_nat_num(F,Na)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_632_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Na: nat,M: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_633_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Na: nat,M: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_634_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_635_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_636_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => aa_num_o(ord_less_num(aa_nat_num(F,Na)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_637_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_638_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_639_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_640_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M4: nat] : Na = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_641_All__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
           => aa_nat_o(P,aa_nat_nat(suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_642_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M5: nat] : Na = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_643_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
            & aa_nat_o(P,aa_nat_nat(suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_644_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_645_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_646_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_Suc_eq_le
tff(fact_647_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_648_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_lessD
tff(fact_649_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_650_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_651_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_eq
tff(fact_652_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_leI
tff(fact_653_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ? [K3: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_imp_Suc_add
tff(fact_654_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ? [K2: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_iff_Suc_add
tff(fact_655_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(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_656_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(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_657_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ~ ! [Q2: nat] : Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ) ).

% less_natE
tff(fact_658_power__gt1,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_659_power__gt1,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_660_power__gt1,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_661_power__gt1,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_662_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_663_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_664_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_665_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_666_power__strict__increasing,axiom,
    ! [Na: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_667_power__strict__increasing,axiom,
    ! [Na: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_668_power__strict__increasing,axiom,
    ! [Na: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_669_power__strict__increasing,axiom,
    ! [Na: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_670_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K3)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K3)) ) ) ) ).

% ex_least_nat_less
tff(fact_671_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_672_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),K)) ) ).

% power_gt_expt
tff(fact_673_zero__less__iff__neq__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_674_gr__implies__not__zero,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_675_not__less__zero,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less_zero
tff(fact_676_gr__zeroI,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr_zeroI
tff(fact_677_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_678_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_679_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_680_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_681_power__divide,axiom,
    ! [A: complex,B: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Na) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Na)),aa_nat_complex(power_power_complex(B),Na)) ).

% power_divide
tff(fact_682_power__divide,axiom,
    ! [A: real,B: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Na) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na)) ).

% power_divide
tff(fact_683_power__divide,axiom,
    ! [A: rat,B: rat,Na: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),Na) = divide_divide_rat(aa_nat_rat(power_power_rat(A),Na),aa_nat_rat(power_power_rat(B),Na)) ).

% power_divide
tff(fact_684_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_685_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_686_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_687_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_688_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_689_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_690_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_691_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_692_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_693_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_694_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_695_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_696_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_697_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_698_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_699_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_700_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_701_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_702_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_703_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_704_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_705_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_strict_mono
tff(fact_706_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_707_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% add_strict_mono
tff(fact_708_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(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_709_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_710_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(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_711_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(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_712_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(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_713_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_714_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(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_715_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(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_716_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(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_717_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_718_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(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_719_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(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_720_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M4: nat] : Na = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_721_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_722_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_723_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_724_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_725_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y2))
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ) ).

% diff_induct
tff(fact_726_nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Na) ) ) ).

% nat_induct
tff(fact_727_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_728_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_729_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_730_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_731_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_732_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_733_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y2),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z2) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_734_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(P,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_735_full__nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% full_nat_induct
tff(fact_736_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_737_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_738_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        | ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_739_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M6)
     => ? [M4: nat] : M6 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_740_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_741_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_742_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_leD
tff(fact_743_infinite__descent0,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Na) ) ) ).

% infinite_descent0
tff(fact_744_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_745_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_746_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_747_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_748_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_749_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_750_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) ).

% add_Suc_shift
tff(fact_751_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc
tff(fact_752_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_753_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_754_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ( M != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% le_neq_implies_less
tff(fact_755_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_or_eq_imp_le
tff(fact_756_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_757_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_imp_le_nat
tff(fact_758_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_759_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% less_add_eq_less
tff(fact_760_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_761_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_762_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_763_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_764_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_765_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(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_766_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_767_power__Suc__less__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_768_power__Suc__less__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_769_power__Suc__less__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_770_power__Suc__less__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_771_power__strict__decreasing,axiom,
    ! [Na: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_772_power__strict__decreasing,axiom,
    ! [Na: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_773_power__strict__decreasing,axiom,
    ! [Na: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_774_power__strict__decreasing,axiom,
    ! [Na: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_775_one__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),Na)) ) ) ).

% one_less_power
tff(fact_776_one__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% one_less_power
tff(fact_777_one__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% one_less_power
tff(fact_778_one__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ) ).

% one_less_power
tff(fact_779_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_780_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_781_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_782_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_783_power__strict__mono,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_784_power__strict__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_785_power__strict__mono,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_786_power__strict__mono,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_787_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_788_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_789_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_790_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_791_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).

% divide_numeral_1
tff(fact_792_divide__numeral__1,axiom,
    ! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).

% divide_numeral_1
tff(fact_793_power__one__over,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Na) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),Na)) ).

% power_one_over
tff(fact_794_power__one__over,axiom,
    ! [A: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),Na) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(A),Na)) ).

% power_one_over
tff(fact_795_power__one__over,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),Na) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),Na)) ).

% power_one_over
tff(fact_796_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Na)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_797_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Na)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_798_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_799_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_800_zero__less__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(Na)) ).

% zero_less_numeral
tff(fact_801_zero__less__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(Na)) ).

% zero_less_numeral
tff(fact_802_zero__less__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_less_numeral
tff(fact_803_zero__less__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_less_numeral
tff(fact_804_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_805_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_806_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% add_less_le_mono
tff(fact_807_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_808_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_809_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_810_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% add_le_less_mono
tff(fact_811_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_812_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_813_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(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_814_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(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_815_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(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_816_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_817_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(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_818_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(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_819_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(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_820_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_821_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_822_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_823_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_824_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_825_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_826_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_827_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_828_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_829_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_830_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_831_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_832_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_833_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_834_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_835_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_836_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_837_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Na)),one_one_real) ).

% not_numeral_less_one
tff(fact_838_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Na)),one_one_rat) ).

% not_numeral_less_one
tff(fact_839_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) ).

% not_numeral_less_one
tff(fact_840_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) ).

% not_numeral_less_one
tff(fact_841_zero__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),Na)) ) ).

% zero_less_power
tff(fact_842_zero__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_less_power
tff(fact_843_zero__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_less_power
tff(fact_844_zero__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_less_power
tff(fact_845_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_846_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_847_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,N3)),aa_nat_num(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_848_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_849_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_850_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_real,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_851_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Na)),aa_nat_set_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_852_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_853_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,Na)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_854_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_855_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Na)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_856_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_real,Na: nat,N3: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,Na)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_857_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_858_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_859_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K3)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K3) ) ) ) ).

% ex_least_nat_le
tff(fact_860_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_861_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_862_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ).

% Suc_eq_plus1
tff(fact_863_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_864_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ).

% Suc_eq_plus1_left
tff(fact_865_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M4)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(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_866_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_867_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_868_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_869_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_870_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_871_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_872_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_873_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_874_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_875_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_876_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_877_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_878_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_879_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_880_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_881_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_882_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_883_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_884_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_885_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_886_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_887_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_888_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_889_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_890_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_891_power__le__imp__le__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_892_power__le__imp__le__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_893_power__le__imp__le__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_894_power__le__imp__le__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na))),aa_nat_real(power_power_real(B),aa_nat_nat(suc,Na)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_895_power__inject__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_896_power__inject__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_897_power__inject__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_898_power__inject__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na)) = aa_nat_real(power_power_real(B),aa_nat_nat(suc,Na)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_899_power__less__imp__less__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_900_power__less__imp__less__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_901_power__less__imp__less__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_902_power__less__imp__less__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_903_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_904_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = zero_zero_nat ) ) ).

% zero_power
tff(fact_905_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(power_power_int(zero_zero_int),Na) = zero_zero_int ) ) ).

% zero_power
tff(fact_906_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(power_power_real(zero_zero_real),Na) = zero_zero_real ) ) ).

% zero_power
tff(fact_907_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = zero_zero_complex ) ) ).

% zero_power
tff(fact_908_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = zero_zero_rat ) ) ).

% zero_power
tff(fact_909_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_910_power__Suc__le__self,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_911_power__Suc__le__self,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_912_power__Suc__le__self,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_913_power__Suc__le__self,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_914_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_915_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_916_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_917_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_918_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_919_power__eq__imp__eq__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_920_power__eq__imp__eq__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_921_power__eq__imp__eq__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_922_power__eq__imp__eq__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),Na) = aa_nat_real(power_power_real(B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_923_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_924_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_925_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_926_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(power_power_real(A),Na) = aa_nat_real(power_power_real(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_927_self__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% self_le_power
tff(fact_928_self__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% self_le_power
tff(fact_929_self__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Na)) ) ) ).

% self_le_power
tff(fact_930_self__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(power_power_real(A),Na)) ) ) ).

% self_le_power
tff(fact_931_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_932_less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% less_exp
tff(fact_933_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,plus_plus_num(V,one))),Na) ).

% Suc_nat_number_of_add
tff(fact_934_div__exp__eq,axiom,
    ! [A: nat,M: nat,Na: 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))),Na)) = 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),Na))) ).

% div_exp_eq
tff(fact_935_div__exp__eq,axiom,
    ! [A: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_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),Na))) ).

% div_exp_eq
tff(fact_936_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_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_937_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(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_938_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(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_939_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_940_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_941_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_942_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_943_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_944_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_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_945_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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_946_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_947_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_948_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(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_949_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(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))) ) ) ) ).

% ex_power_ivl2
tff(fact_950_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(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))) ) ) ) ).

% ex_power_ivl1
tff(fact_951_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_952_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_953_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_954_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_955_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_956_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_957_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_958_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_959_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_960_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_961_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_962_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_963_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_964_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_965_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_966_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_967_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_968_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_969_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_970_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_971_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_972_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_973_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_974_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_975_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_976_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_977_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_978_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_979_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_980_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_981_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_982_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_983_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_984_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_985_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_986_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(78)
tff(fact_987_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_num(M),one) ).

% semiring_norm(75)
tff(fact_988_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_989_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_990_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_991_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_992_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_993_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = divide_divide_rat(B,A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_994_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_995_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( divide_divide_rat(B,A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_996_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_997_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_998_divide__self__if,axiom,
    ! [A: rat] :
      divide_divide_rat(A,A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_999_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_1000_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_1001_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_1002_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_1003_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_1004_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = divide_divide_rat(A,B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1005_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_1006_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_1007_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1008_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_1009_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat ) ) ).

% div_less
tff(fact_1010_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_1011_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(ord_less_num(one),aa_num_num(bit0,Na)) ).

% semiring_norm(76)
tff(fact_1012_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(2)
tff(fact_1013_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_1014_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_1015_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_1016_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_1017_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_1018_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_1019_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1020_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1021_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_1022_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_1023_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_1024_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_1025_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1026_iadd__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,Na) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( Na = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_1027_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_1028_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_1029_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(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)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_1030_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(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_1031_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_1032_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1033_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(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_1034_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_1) ).

% linordered_field_no_ub
tff(fact_1035_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_1) ).

% linordered_field_no_ub
tff(fact_1036_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X3) ).

% linordered_field_no_lb
tff(fact_1037_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X3) ).

% linordered_field_no_lb
tff(fact_1038_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_1039_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_1040_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ).

% add_divide_distrib
tff(fact_1041_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(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,Na),K)) ) ).

% div_le_mono
tff(fact_1042_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ).

% div_le_dividend
tff(fact_1043_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_1044_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_1045_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_right_mono
tff(fact_1046_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_1047_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_1048_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_1049_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_1050_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_1051_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_1052_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_1053_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_1054_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_1055_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_1056_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_1057_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C)) ) ) ).

% divide_right_mono_neg
tff(fact_1058_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(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_1059_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_neg_neg
tff(fact_1060_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(X,Y)) ) ) ).

% divide_neg_neg
tff(fact_1061_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_1062_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(X,Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_1063_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_1064_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(X,Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_1065_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_pos_pos
tff(fact_1066_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(X,Y)) ) ) ).

% divide_pos_pos
tff(fact_1067_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_1068_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_1069_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_1070_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_1071_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_1072_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_1073_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_1074_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono
tff(fact_1075_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(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_1076_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1077_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_1078_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_1079_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat(A,B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_1080_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1081_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_1082_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_epsilon
tff(fact_1083_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% field_le_epsilon
tff(fact_1084_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_le
tff(fact_1085_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(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_1086_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less
tff(fact_1087_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(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_1088_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less2
tff(fact_1089_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(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_1090_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_1091_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_1092_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_1093_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_1094_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y)) ) ) ).

% divide_nonneg_pos
tff(fact_1095_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_1096_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y)) ) ) ).

% divide_nonpos_neg
tff(fact_1097_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_1098_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_1099_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_1100_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_1101_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_1102_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_1103_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_1104_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_1105_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_1106_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_1107_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_1108_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_1109_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_1110_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_1111_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_1112_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Na) ).

% numeral_Bit0_div_2
tff(fact_1113_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Na) ).

% numeral_Bit0_div_2
tff(fact_1114_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_1115_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_1116_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = M )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_1117_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_1118_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_1119_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_1120_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_1121_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_1122_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_1123_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1124_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Na)),Na) ) ).

% buildup_gives_valid
tff(fact_1125_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_1126_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_1127_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_1128_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_1129_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_1130_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1131_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1132_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1133_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_1134_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_1135_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_1136_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_1137_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_1138_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_1139_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_1140_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,numeral_numeral_rat(aa_num_num(bit0,one)))),divide_divide_rat(X,numeral_numeral_rat(aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_1141_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1142_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_1143_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_1144_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_1145_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1146_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1147_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_1148_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1149_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_1150_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1151_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Na)),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_1152_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1153_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1154_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1155_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1156_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( member_o((X),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1157_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1158_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1159_high__def,axiom,
    ! [X: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na) = 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))),Na)) ).

% high_def
tff(fact_1160_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N)
             => aa_Extended_enat_o(P,M2) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Na) ) ).

% enat_less_induct
tff(fact_1161_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1162_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1163_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1164_realpow__pos__nth2,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
          & ( aa_nat_real(power_power_real(R2),aa_nat_nat(suc,Na)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_1165_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B2)
    <=> ! [X4: complex] :
          ( member_complex(X4,set_complex2(Xs))
         => member_complex(X4,B2) ) ) ).

% subset_code(1)
tff(fact_1166_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B2)
    <=> ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => member_real(X4,B2) ) ) ).

% subset_code(1)
tff(fact_1167_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B2)
    <=> ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => member_set_nat(X4,B2) ) ) ).

% subset_code(1)
tff(fact_1168_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X4,B2) ) ) ).

% subset_code(1)
tff(fact_1169_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B2)
    <=> ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => member_int(X4,B2) ) ) ).

% subset_code(1)
tff(fact_1170_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B2)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X4),B2) ) ) ).

% subset_code(1)
tff(fact_1171_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_1172_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1173_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1174_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1175_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1176_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1177_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1178_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Na ).

% Ex_list_of_length
tff(fact_1179_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Na ).

% Ex_list_of_length
tff(fact_1180_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Na ).

% Ex_list_of_length
tff(fact_1181_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Na ).

% Ex_list_of_length
tff(fact_1182_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D2) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1183_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D2)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D2) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1184_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1185_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1186_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1187_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1188_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1189_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),Na)) ).

% two_realpow_ge_one
tff(fact_1190_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
            & ( aa_nat_real(power_power_real(X2),Na) = A )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
                  & ( aa_nat_real(power_power_real(Y3),Na) = A ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1191_realpow__pos__nth,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ( aa_nat_real(power_power_real(R2),Na) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_1192_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_1193_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_1194_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_1195_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_1196_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B)) ) ) ).

% zdiv_mono1
tff(fact_1197_zdiv__mono2,axiom,
    ! [A: int,B3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3)) ) ) ) ).

% zdiv_mono2
tff(fact_1198_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1199_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_1200_zdiv__mono2__neg,axiom,
    ! [A: int,B3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1201_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_1202_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_1203_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_1204_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_1205_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1206_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1207_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1208_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1209_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1210_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na)),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_1211_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_1212_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_1213_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_1214_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_1215_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1216_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1217_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1218_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1219_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_1220_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_1221_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_1222_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_1223_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_1224_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_1225_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_1226_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_1227_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_1228_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_1229_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_1230_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_1231_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1232_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1233_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1234_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_1235_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_1236_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_1237_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_1238_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_1239_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_1240_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_1241_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_1242_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_1243_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_1244_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_1245_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1246_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_1247_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_1248_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(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_1249_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_1250_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(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_1251_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(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_1252_high__inv,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( 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))),Na))),X)),Na) = Y ) ) ).

% high_inv
tff(fact_1253_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,X: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na)),X) ).

% buildup_nothing_in_leaf
tff(fact_1254_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1255_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1256_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1257_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_1258_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1259_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1260_even__succ__div__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_1261_even__succ__div__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( 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))),Na)) = 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))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_1262_even__succ__div__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(aa_int_fun_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))),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_1263_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_1264_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_1265_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1266_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_1267_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1268_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_1269_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_1270_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1271_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_1272_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1273_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_1274_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_1275_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_1276_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_1277_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_1278_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_1279_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_1280_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1281_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_1282_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_1283_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_1284_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_1285_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1286_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_1287_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_1288_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_1289_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_1290_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(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1291_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1292_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1293_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,times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1294_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,times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1295_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_1296_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),numeral_numeral_real(Na)) = numeral_numeral_real(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_1297_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(Na)) = numeral_numeral_rat(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_1298_numeral__times__numeral,axiom,
    ! [M: num,Na: 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,Na)) = aa_num_nat(numeral_numeral_nat,times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_1299_numeral__times__numeral,axiom,
    ! [M: num,Na: 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,Na)) = aa_num_int(numeral_numeral_int,times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_1300_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_1301_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_1302_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1303_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_1304_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_1305_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_1306_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_1307_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1308_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_1309_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_1310_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_1311_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_1312_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),B),C) ).

% times_divide_eq_right
tff(fact_1313_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_1314_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_1315_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),C),B) ).

% divide_divide_eq_right
tff(fact_1316_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_1317_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_1318_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_1319_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_1320_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_1321_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(times_times_rat(B),A),C) ).

% times_divide_eq_left
tff(fact_1322_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_1323_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(A),zero_zero_complex) ).

% dvd_0_right
tff(fact_1324_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_1325_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_1326_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_1327_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_1328_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_1329_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(dvd_dvd_complex(zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_1330_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_1331_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_1332_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_1333_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_1334_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1335_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1336_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1337_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1338_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1339_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1340_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1341_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1342_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1343_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1344_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_1345_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1346_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_1347_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_1348_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_1349_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1350_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1351_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_1352_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1353_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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,Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1354_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1355_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1356_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_1357_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_1358_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_1359_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_1360_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_1361_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_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_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_1375_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_1376_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_1377_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_1378_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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_1379_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_1380_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1381_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: 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)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1382_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A,B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1383_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_1384_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_1385_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1386_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_1387_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_1388_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1389_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_1390_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_1391_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1392_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_1393_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_1394_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1395_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_1396_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_1397_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_1398_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_1399_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: 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)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_1400_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_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)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_1401_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_1402_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_1403_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1404_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_1405_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1406_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_1407_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_1408_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1409_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_1410_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1411_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_1412_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_1413_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B)),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_1414_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_1415_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_1416_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_1417_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_1418_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),numeral_numeral_rat(V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_1419_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_1420_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_1421_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_1422_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_1423_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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 )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_1424_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_1425_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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 )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_1426_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1427_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1428_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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 )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1429_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1430_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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 )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1431_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1432_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1433_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1434_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1435_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1436_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1437_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_1438_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_1439_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_1440_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1441_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1442_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1443_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1444_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1445_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1446_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1447_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(times_times_rat(C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1448_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1449_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1450_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_1451_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_1452_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_1453_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_1454_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_1455_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_1456_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_1457_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_1458_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_1459_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_1460_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_1461_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_1462_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_1463_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_1464_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_1465_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_1466_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_1467_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_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,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_1468_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1469_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1470_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% mult_less_cancel2
tff(fact_1471_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_1472_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1473_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc_right
tff(fact_1474_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_1475_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_1476_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_1477_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_1478_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_1479_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1480_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1481_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1482_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1483_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1484_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1485_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1486_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1487_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1488_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1489_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1490_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1491_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1492_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1493_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_1494_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_1495_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1496_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_1497_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_1498_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1499_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_1500_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_1501_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_1502_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_1503_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_1504_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_1505_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_1506_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_1507_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_1508_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_1509_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_1510_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_1511_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_1512_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_1513_even__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc
tff(fact_1514_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_1515_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ) ) ).

% one_le_mult_iff
tff(fact_1516_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% mult_le_cancel2
tff(fact_1517_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1518_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_1519_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1520_power__add__numeral,axiom,
    ! [A: complex,M: num,Na: 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,Na))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na))) ).

% power_add_numeral
tff(fact_1521_power__add__numeral,axiom,
    ! [A: real,M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,Na))) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na))) ).

% power_add_numeral
tff(fact_1522_power__add__numeral,axiom,
    ! [A: rat,M: num,Na: 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,Na))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na))) ).

% power_add_numeral
tff(fact_1523_power__add__numeral,axiom,
    ! [A: nat,M: num,Na: 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,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na))) ).

% power_add_numeral
tff(fact_1524_power__add__numeral,axiom,
    ! [A: int,M: num,Na: 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,Na))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na))) ).

% power_add_numeral
tff(fact_1525_power__add__numeral2,axiom,
    ! [A: complex,M: num,Na: 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,Na))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na)))),B) ).

% power_add_numeral2
tff(fact_1526_power__add__numeral2,axiom,
    ! [A: real,M: num,Na: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,Na))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na)))),B) ).

% power_add_numeral2
tff(fact_1527_power__add__numeral2,axiom,
    ! [A: rat,M: num,Na: 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,Na))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,plus_plus_num(M,Na)))),B) ).

% power_add_numeral2
tff(fact_1528_power__add__numeral2,axiom,
    ! [A: nat,M: num,Na: 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,Na))),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,plus_plus_num(M,Na)))),B) ).

% power_add_numeral2
tff(fact_1529_power__add__numeral2,axiom,
    ! [A: int,M: num,Na: 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,Na))),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,plus_plus_num(M,Na)))),B) ).

% power_add_numeral2
tff(fact_1530_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_1531_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_1532_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_1533_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_1534_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_1535_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(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))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_1536_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_1537_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_1538_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_1539_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_1540_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_1541_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1542_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1543_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1544_power__less__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Na)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_1545_power__less__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_1546_power__less__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_1547_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1548_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1549_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1550_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_1551_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_1552_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_1553_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1554_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(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 )
        | ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1555_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(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 )
        | ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1556_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1557_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1558_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_1559_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1560_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(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_1561_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_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_1562_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1563_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1564_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_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(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_1565_even__power,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_1566_even__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_1567_even__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_1568_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1569_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(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_1570_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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_1571_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(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_1572_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(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_1573_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(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_1574_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(power_power_real(X),N)) ) ).

% real_arch_pow
tff(fact_1575_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_1576_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1577_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvdE
tff(fact_1578_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K3: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvdE
tff(fact_1579_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K3: rat] : A != aa_rat_rat(times_times_rat(B),K3) ) ).

% dvdE
tff(fact_1580_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvdE
tff(fact_1581_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvdE
tff(fact_1582_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_1583_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_1584_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_1585_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_1586_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_1587_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K2: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvd_def
tff(fact_1588_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K2: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvd_def
tff(fact_1589_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K2: rat] : A = aa_rat_rat(times_times_rat(B),K2) ) ).

% dvd_def
tff(fact_1590_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K2: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvd_def
tff(fact_1591_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K2: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvd_def
tff(fact_1592_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_1593_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_1594_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_1595_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_1596_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_1597_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_1598_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_1599_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_1600_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_1601_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_1602_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_1603_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_1604_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_1605_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_1606_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_1607_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_1608_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_1609_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_1610_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_1611_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_1612_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_1613_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_1614_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_1615_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).

% dvd_triv_left
tff(fact_1616_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_1617_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_1618_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_1619_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_1620_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_1621_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_1622_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% mult_dvd_mono
tff(fact_1623_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_1624_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_1625_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_1626_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_1627_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_1628_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_1629_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_1630_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).

% dvd_triv_right
tff(fact_1631_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_1632_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_1633_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X3: real] : member_real(X3,S2)
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( member_real(X2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z3) )
       => ? [Y2: real] :
            ( ! [X3: real] :
                ( member_real(X3,S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y2) )
            & ! [Z3: real] :
                ( ! [X2: real] :
                    ( member_real(X2,S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z3) ) ) ) ) ).

% complete_real
tff(fact_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_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_1642_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_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_1651_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_1652_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_1653_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_1654_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_1655_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_1656_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_1657_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_1658_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_1659_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1660_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1661_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1662_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1663_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1664_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1665_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_1666_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_1667_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_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) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_1668_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_1669_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_1670_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_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) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_1671_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1672_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1673_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1674_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_1675_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_1676_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_1677_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_1678_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_1679_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_1680_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_1681_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_1682_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_1683_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_1684_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_1685_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_1686_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_1687_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_1688_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_1689_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).

% unit_dvdE
tff(fact_1690_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_1691_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_1692_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1693_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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_1694_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_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_1695_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1696_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1697_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1698_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1699_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1700_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1701_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1702_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1703_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1704_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1705_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1706_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1707_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_1708_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_1709_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_1710_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_1711_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_1712_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_1713_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_1714_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_1715_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_1716_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_1717_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_1718_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_1719_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_1720_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_1721_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_1722_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1723_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_1724_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_1725_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_1726_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_1727_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_1728_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(dvd_dvd_complex(zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_1729_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_1730_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_1731_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_1732_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_1733_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_1734_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_1735_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_1736_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_1737_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_1738_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_1739_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_1740_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(one_one_complex),A) ).

% one_dvd
tff(fact_1741_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_1742_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_1743_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_1744_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_1745_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_1746_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_1747_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ) ).

% dvd_add
tff(fact_1748_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_1749_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_1750_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_1751_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_1752_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_1753_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_1754_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_1755_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1756_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1757_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1758_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1759_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1760_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_1761_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_1762_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_1763_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1764_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( aa_complex_o(dvd_dvd_complex(C),A)
       => ( aa_complex_o(dvd_dvd_complex(C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1765_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1766_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1767_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1768_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1769_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1770_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(C),A)
     => ( aa_complex_o(dvd_dvd_complex(C),B)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1771_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1772_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1773_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1774_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1775_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Na)) ) ).

% dvd_power_same
tff(fact_1776_dvd__power__same,axiom,
    ! [X: nat,Y: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Na)) ) ).

% dvd_power_same
tff(fact_1777_dvd__power__same,axiom,
    ! [X: int,Y: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),Na)),aa_nat_int(power_power_int(Y),Na)) ) ).

% dvd_power_same
tff(fact_1778_dvd__power__same,axiom,
    ! [X: real,Y: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(X),Na)),aa_nat_real(power_power_real(Y),Na)) ) ).

% dvd_power_same
tff(fact_1779_dvd__power__same,axiom,
    ! [X: complex,Y: complex,Na: nat] :
      ( aa_complex_o(dvd_dvd_complex(X),Y)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(X),Na)),aa_nat_complex(power_power_complex(Y),Na)) ) ).

% dvd_power_same
tff(fact_1780_dvd__power__same,axiom,
    ! [X: rat,Y: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(X),Na)),aa_nat_rat(power_power_rat(Y),Na)) ) ).

% dvd_power_same
tff(fact_1781_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_1782_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_1783_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_1784_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_1785_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_1786_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_1787_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_1788_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_1789_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_1790_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_1791_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_1792_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_1793_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_1794_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_1795_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_1796_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1797_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_1798_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_1799_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_1800_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_1801_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1802_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_1803_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_1804_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_1805_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_1806_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1807_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_1808_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1809_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_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1817_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1818_ring__class_Oring__distribs_I2_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(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_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)) ).

% ring_class.ring_distribs(2)
tff(fact_1819_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1820_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1821_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),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,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1822_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1823_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1824_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1825_comm__semiring__class_Odistrib,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(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_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)) ).

% comm_semiring_class.distrib
tff(fact_1826_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_1827_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_1828_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_1829_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),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,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_1830_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_1831_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_1832_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_1833_distrib__right,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(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_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)) ).

% distrib_right
tff(fact_1834_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1835_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1836_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1837_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: 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)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),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(plus_plus_int,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1838_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_1839_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_1840_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_1841_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1842_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_1843_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(X,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),W),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1844_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_1845_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_1846_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(X,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_1847_power__commutes,axiom,
    ! [A: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Na)) ).

% power_commutes
tff(fact_1848_power__commutes,axiom,
    ! [A: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),Na)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),Na)) ).

% power_commutes
tff(fact_1849_power__commutes,axiom,
    ! [A: rat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Na)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na)) ).

% power_commutes
tff(fact_1850_power__commutes,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),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),Na)) ).

% power_commutes
tff(fact_1851_power__commutes,axiom,
    ! [A: int,Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na)) ).

% power_commutes
tff(fact_1852_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),aa_nat_complex(power_power_complex(B),Na)) ).

% power_mult_distrib
tff(fact_1853_power__mult__distrib,axiom,
    ! [A: real,B: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na)) ).

% power_mult_distrib
tff(fact_1854_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ).

% power_mult_distrib
tff(fact_1855_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Na: 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)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ).

% power_mult_distrib
tff(fact_1856_power__mult__distrib,axiom,
    ! [A: int,B: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ).

% power_mult_distrib
tff(fact_1857_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,Na: 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),Na)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(X),Na)) ) ) ).

% power_commuting_commutes
tff(fact_1858_power__commuting__commutes,axiom,
    ! [X: real,Y: real,Na: 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(power_power_real(X),Na)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(power_power_real(X),Na)) ) ) ).

% power_commuting_commutes
tff(fact_1859_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,Na: 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),Na)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(X),Na)) ) ) ).

% power_commuting_commutes
tff(fact_1860_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,Na: 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),Na)),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),Na)) ) ) ).

% power_commuting_commutes
tff(fact_1861_power__commuting__commutes,axiom,
    ! [X: int,Y: int,Na: 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),Na)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),Na)) ) ) ).

% power_commuting_commutes
tff(fact_1862_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_1863_power__mult,axiom,
    ! [A: nat,M: nat,Na: 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),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),Na) ).

% power_mult
tff(fact_1864_power__mult,axiom,
    ! [A: int,M: nat,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),Na) ).

% power_mult
tff(fact_1865_power__mult,axiom,
    ! [A: real,M: nat,Na: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),M)),Na) ).

% power_mult
tff(fact_1866_power__mult,axiom,
    ! [A: complex,M: nat,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),Na) ).

% power_mult
tff(fact_1867_power__mult,axiom,
    ! [A: rat,M: nat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),M)),Na) ).

% power_mult
tff(fact_1868_mult__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ).

% mult_0
tff(fact_1869_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1870_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(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_1871_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_1872_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(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_1873_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_1874_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(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_1875_add__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),K)) ).

% add_mult_distrib
tff(fact_1876_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ).

% add_mult_distrib2
tff(fact_1877_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_1878_nat__mult__1__right,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ).

% nat_mult_1_right
tff(fact_1879_nat__mult__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ).

% nat_mult_1
tff(fact_1880_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Q3) ).

% div_mult2_eq
tff(fact_1881_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B4),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B4 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B4) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B4) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1882_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B4),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1883_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B4),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B4) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1884_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1885_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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_1886_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1887_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1888_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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_1889_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1890_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_1891_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_1892_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B4: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_1893_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_1894_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_1895_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1896_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1897_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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_1898_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_1899_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_1900_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_1901_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1902_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(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_1903_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_1904_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1905_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_1906_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1907_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_1908_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_1909_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_1910_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_1911_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_1912_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_1913_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_1914_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_1915_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_1916_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1917_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1918_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_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,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1919_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1920_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1921_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_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,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1922_div__power,axiom,
    ! [B: code_integer,A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Na) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% div_power
tff(fact_1923_div__power,axiom,
    ! [B: nat,A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ).

% div_power
tff(fact_1924_div__power,axiom,
    ! [B: int,A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Na) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% div_power
tff(fact_1925_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1926_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1927_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ) ) ).

% mult_mono
tff(fact_1928_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1929_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1930_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1931_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1932_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1933_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_1934_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(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_1935_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_1936_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_1937_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(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_1938_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_1939_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(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_1940_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_1941_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(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_1942_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1943_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_1944_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1945_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(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_1946_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(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_1947_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(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_1948_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_1949_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(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_1950_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_1951_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(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_1952_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(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_1953_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(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_1954_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(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_1955_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_1956_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1957_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1958_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1959_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_1960_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(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_1961_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(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_1962_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_1963_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1964_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(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_1965_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(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_1966_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1967_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1968_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(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_1969_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_1970_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_1971_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1972_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(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_1973_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(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_1974_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_1975_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1976_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(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_1977_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_1978_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1979_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_1980_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_1981_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_1982_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(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_1983_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(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_1984_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(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_1985_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_1986_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1987_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1988_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1989_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_1990_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1991_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(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_1992_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1993_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1994_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1995_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1996_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1997_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1998_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1999_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2000_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2001_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2002_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2003_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2004_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_2005_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2006_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2007_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2008_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2009_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2010_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2011_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2012_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2013_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2014_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_2015_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2016_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2017_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_2018_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2019_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2020_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2021_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2022_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2023_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2024_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2025_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2026_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2027_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2028_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(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_2029_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(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_2030_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_2031_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_2032_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(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_2033_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(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_2034_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(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_2035_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2036_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2037_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2038_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(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_2039_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(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_2040_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(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_2041_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_2042_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(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_2043_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(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_2044_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(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_2045_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2046_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2047_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2048_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(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_2049_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(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_2050_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(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_2051_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(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_2052_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,Na: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2053_dvd__power__le,axiom,
    ! [X: nat,Y: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_2054_dvd__power__le,axiom,
    ! [X: int,Y: int,Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),Na)),aa_nat_int(power_power_int(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2055_dvd__power__le,axiom,
    ! [X: real,Y: real,Na: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(X),Na)),aa_nat_real(power_power_real(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2056_dvd__power__le,axiom,
    ! [X: complex,Y: complex,Na: nat,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(X),Na)),aa_nat_complex(power_power_complex(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2057_dvd__power__le,axiom,
    ! [X: rat,Y: rat,Na: nat,M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(X),Na)),aa_nat_rat(power_power_rat(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2058_power__le__dvd,axiom,
    ! [A: code_integer,Na: nat,B: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2059_power__le__dvd,axiom,
    ! [A: nat,Na: nat,B: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2060_power__le__dvd,axiom,
    ! [A: int,Na: nat,B: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2061_power__le__dvd,axiom,
    ! [A: real,Na: nat,B: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2062_power__le__dvd,axiom,
    ! [A: complex,Na: nat,B: complex,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2063_power__le__dvd,axiom,
    ! [A: rat,Na: nat,B: rat,M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),Na)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2064_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_2065_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(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),Na)) ) ).

% le_imp_power_dvd
tff(fact_2066_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_2067_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_2068_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_2069_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na)) ) ).

% le_imp_power_dvd
tff(fact_2070_less__1__mult,axiom,
    ! [M: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),Na)) ) ) ).

% less_1_mult
tff(fact_2071_less__1__mult,axiom,
    ! [M: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),Na)) ) ) ).

% less_1_mult
tff(fact_2072_less__1__mult,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% less_1_mult
tff(fact_2073_less__1__mult,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)) ) ) ).

% less_1_mult
tff(fact_2074_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_2075_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_2076_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(X,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2077_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_2078_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2079_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2080_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_2081_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2082_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2083_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_2084_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_2085_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2086_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_2087_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_2088_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_2089_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_2090_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_2091_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2092_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_2093_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_2094_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2095_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_2096_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).

% mult_numeral_1
tff(fact_2097_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_2098_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_2099_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_2100_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_2101_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).

% mult_numeral_1_right
tff(fact_2102_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_2103_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_2104_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_2105_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_2106_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,Na: 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),Na)),aa_nat_complex(power_power_complex(Y),Na)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_2107_left__right__inverse__power,axiom,
    ! [X: real,Y: real,Na: 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(power_power_real(X),Na)),aa_nat_real(power_power_real(Y),Na)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_2108_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,Na: 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),Na)),aa_nat_rat(power_power_rat(Y),Na)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2109_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,Na: 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),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Na)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2110_left__right__inverse__power,axiom,
    ! [X: int,Y: int,Na: 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),Na)),aa_nat_int(power_power_int(Y),Na)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2111_power__Suc2,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),A) ).

% power_Suc2
tff(fact_2112_power__Suc2,axiom,
    ! [A: real,Na: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),Na)),A) ).

% power_Suc2
tff(fact_2113_power__Suc2,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Na)),A) ).

% power_Suc2
tff(fact_2114_power__Suc2,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),A) ).

% power_Suc2
tff(fact_2115_power__Suc2,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),A) ).

% power_Suc2
tff(fact_2116_power__Suc,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Na)) ).

% power_Suc
tff(fact_2117_power__Suc,axiom,
    ! [A: real,Na: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),Na)) ).

% power_Suc
tff(fact_2118_power__Suc,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na)) ).

% power_Suc
tff(fact_2119_power__Suc,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ).

% power_Suc
tff(fact_2120_power__Suc,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na)) ).

% power_Suc
tff(fact_2121_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2122_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_2123_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_2124_power__add,axiom,
    ! [A: complex,M: nat,Na: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Na)) ).

% power_add
tff(fact_2125_power__add,axiom,
    ! [A: real,M: nat,Na: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),Na)) ).

% power_add
tff(fact_2126_power__add,axiom,
    ! [A: rat,M: nat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na)) ).

% power_add
tff(fact_2127_power__add,axiom,
    ! [A: nat,M: nat,Na: 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),Na)) = 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),Na)) ).

% power_add
tff(fact_2128_power__add,axiom,
    ! [A: int,M: nat,Na: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = 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),Na)) ).

% power_add
tff(fact_2129_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(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_2130_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(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_2131_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2132_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_2133_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_2134_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc
tff(fact_2135_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_2136_dbl__def,axiom,
    ! [X: rat] : neg_numeral_dbl_rat(X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X) ).

% dbl_def
tff(fact_2137_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_2138_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
     => ( ( Na = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2139_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_2140_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_2141_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(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),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_2142_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B4),one_one_Code_integer) ) ).

% oddE
tff(fact_2143_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4)),one_one_nat) ) ).

% oddE
tff(fact_2144_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B4: 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))),B4)),one_one_int) ) ).

% oddE
tff(fact_2145_power__odd__eq,axiom,
    ! [A: complex,Na: 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))),Na))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2146_power__odd__eq,axiom,
    ! [A: real,Na: nat] : aa_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))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2147_power__odd__eq,axiom,
    ! [A: rat,Na: 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))),Na))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2148_power__odd__eq,axiom,
    ! [A: nat,Na: 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))),Na))) = 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),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2149_power__odd__eq,axiom,
    ! [A: int,Na: 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))),Na))) = 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),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2150_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2151_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2152_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2153_even__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,Na))) ).

% even_numeral
tff(fact_2154_even__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ).

% even_numeral
tff(fact_2155_even__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) ).

% even_numeral
tff(fact_2156_is__unit__power__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_2157_is__unit__power__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_2158_is__unit__power__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_2159_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2160_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2161_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2162_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2163_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2164_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2165_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2166_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2167_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2168_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2169_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2170_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2171_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2172_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2173_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2174_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2175_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2176_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2177_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2178_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2179_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2180_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2181_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2182_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2183_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2184_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2185_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2186_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2187_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2188_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2189_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2190_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2191_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2192_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2193_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2194_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2195_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2196_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2197_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2198_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2199_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2200_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2201_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2202_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2203_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2204_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2205_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2206_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2207_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2208_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2209_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2210_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(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_2211_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2212_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2213_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(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_2214_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2215_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2216_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(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_2217_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(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_2218_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2219_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2220_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2221_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2222_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2223_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_2224_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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_2225_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_2226_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_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_2227_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(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_2228_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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_2229_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_2230_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_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_2231_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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_2232_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(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_2233_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_2234_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(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_2235_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2236_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2237_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_2238_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2239_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2240_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2241_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2242_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2243_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2244_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2245_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2246_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2247_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2248_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2249_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2250_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(X,Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2251_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2252_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),divide_divide_rat(X,Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2253_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(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_2254_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2255_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(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_2256_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2257_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2258_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C),numeral_numeral_real(W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2259_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2260_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2261_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B,numeral_numeral_real(W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2262_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2263_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,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_2264_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,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_2265_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2266_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,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_2267_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,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_2268_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),Z)),B),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2269_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_2270_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_2271_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_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_2272_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_2273_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_2274_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,Y)),Z) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2275_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_2276_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_2277_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(X,Y)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2278_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_2279_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_2280_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(Y,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),Z)),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2281_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_2282_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_2283_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,Z)),Y) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2284_power__less__power__Suc,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(A),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),Na))) ) ).

% power_less_power_Suc
tff(fact_2285_power__less__power__Suc,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_less_power_Suc
tff(fact_2286_power__less__power__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_less_power_Suc
tff(fact_2287_power__less__power__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_less_power_Suc
tff(fact_2288_power__gt1__lemma,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),Na))) ) ).

% power_gt1_lemma
tff(fact_2289_power__gt1__lemma,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_gt1_lemma
tff(fact_2290_power__gt1__lemma,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(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),Na))) ) ).

% power_gt1_lemma
tff(fact_2291_power__gt1__lemma,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(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),Na))) ) ).

% power_gt1_lemma
tff(fact_2292_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_2293_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_2294_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_2295_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_2296_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% one_less_mult
tff(fact_2297_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_2298_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_2299_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_2300_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_2301_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,X: nat,M7: nat] :
      ( aa_nat_o(P,X)
     => ( ! [X2: nat] :
            ( aa_nat_o(P,X2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M7) )
       => ~ ! [M4: nat] :
              ( aa_nat_o(P,M4)
             => ~ ! [X3: nat] :
                    ( aa_nat_o(P,X3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M4) ) ) ) ) ).

% bounded_Max_nat
tff(fact_2302_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_2303_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_2304_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_2305_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_2306_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_2307_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_2308_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_2309_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(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_2310_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(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_2311_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_2312_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(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_2313_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2314_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,Na: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),Na))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_2315_dvd__power__iff,axiom,
    ! [X: nat,M: nat,Na: nat] :
      ( ( X != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_2316_dvd__power__iff,axiom,
    ! [X: int,M: nat,Na: nat] :
      ( ( X != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),Na))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_2317_dvd__power,axiom,
    ! [Na: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),Na)) ) ).

% dvd_power
tff(fact_2318_dvd__power,axiom,
    ! [Na: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Na)) ) ).

% dvd_power
tff(fact_2319_dvd__power,axiom,
    ! [Na: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(power_power_int(X),Na)) ) ).

% dvd_power
tff(fact_2320_dvd__power,axiom,
    ! [Na: nat,X: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_nat_real(power_power_real(X),Na)) ) ).

% dvd_power
tff(fact_2321_dvd__power,axiom,
    ! [Na: nat,X: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(X),aa_nat_complex(power_power_complex(X),Na)) ) ).

% dvd_power
tff(fact_2322_dvd__power,axiom,
    ! [Na: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( X = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),Na)) ) ).

% dvd_power
tff(fact_2323_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_2324_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),X)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_2325_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2326_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2327_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2328_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2329_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2330_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2331_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2332_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2333_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2334_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2335_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2336_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2337_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2338_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2339_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2340_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2341_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2342_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2343_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2344_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2345_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2346_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2347_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2348_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2349_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2350_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(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_2351_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),divide_divide_rat(X,Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2352_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2353_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2354_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2355_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2356_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2357_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2358_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2359_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2360_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2361_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2362_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2363_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono
tff(fact_2364_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(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_2365_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2366_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2367_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_2368_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_2369_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_2370_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(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_2371_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(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_2372_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2373_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2374_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2375_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2376_power__Suc__less,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),Na))),aa_nat_real(power_power_real(A),Na)) ) ) ).

% power_Suc_less
tff(fact_2377_power__Suc__less,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Na))),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_Suc_less
tff(fact_2378_power__Suc__less,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(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),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% power_Suc_less
tff(fact_2379_power__Suc__less,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))),aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_Suc_less
tff(fact_2380_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_2381_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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_2382_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_2383_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_2384_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_2385_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_2386_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_2387_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_2388_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_2389_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_2390_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_2391_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2392_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2393_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_2394_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_2395_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_2396_power4__eq__xxxx,axiom,
    ! [X: real] : aa_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_2397_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_2398_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_2399_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_2400_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_2401_power2__eq__square,axiom,
    ! [A: real] : aa_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_2402_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_2403_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_2404_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_2405_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: 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))),Na) ).

% Suc_double_not_eq_double
tff(fact_2406_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: 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))),Na)) ).

% double_not_eq_Suc_double
tff(fact_2407_power__even__eq,axiom,
    ! [A: nat,Na: 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))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2408_power__even__eq,axiom,
    ! [A: int,Na: 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))),Na)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2409_power__even__eq,axiom,
    ! [A: real,Na: nat] : aa_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))),Na)) = aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2410_power__even__eq,axiom,
    ! [A: complex,Na: 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))),Na)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2411_power__even__eq,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2412_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_2413_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_2414_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2415_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I3)),J3) )
               => aa_nat_o(P,I3) ) ) ) ) ) ).

% split_div
tff(fact_2416_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_2417_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)))) ) ).

% dividend_less_times_div
tff(fact_2418_power__mono__odd,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono_odd
tff(fact_2419_power__mono__odd,axiom,
    ! [Na: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono_odd
tff(fact_2420_power__mono__odd,axiom,
    ! [Na: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na)) ) ) ).

% power_mono_odd
tff(fact_2421_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_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_2422_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(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_2423_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(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_2424_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2425_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2426_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2427_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2428_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_2429_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_2430_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_2431_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_2432_zero__le__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_2433_zero__le__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_2434_zero__le__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_2435_zero__le__odd__power,axiom,
    ! [Na: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_2436_zero__le__odd__power,axiom,
    ! [Na: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_2437_zero__le__odd__power,axiom,
    ! [Na: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Na))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_2438_zero__le__even__power,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_even_power
tff(fact_2439_zero__le__even__power,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_even_power
tff(fact_2440_zero__le__even__power,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),Na)) ) ).

% zero_le_even_power
tff(fact_2441_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_2442_power2__sum,axiom,
    ! [X: real,Y: real] : aa_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_2443_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_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(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_2444_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_2445_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_2446_zero__le__even__power_H,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(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))),Na))) ).

% zero_le_even_power'
tff(fact_2447_zero__le__even__power_H,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(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))),Na))) ).

% zero_le_even_power'
tff(fact_2448_zero__le__even__power_H,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_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))),Na))) ).

% zero_le_even_power'
tff(fact_2449_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(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))),N))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_bit_induct
tff(fact_2450_zero__less__power__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_2451_zero__less__power__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_2452_zero__less__power__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_2453_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(aa_rat_fun_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_2454_sum__squares__bound,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2455_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))),Na))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2456_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))),Na))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2457_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_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))),Na))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2458_odd__power__less__zero,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_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))),Na)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_2459_odd__power__less__zero,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(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))),Na)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_2460_odd__power__less__zero,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(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))),Na)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_2461_power__le__zero__eq,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2462_power__le__zero__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2463_power__le__zero__eq,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2464_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) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2465_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_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) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2466_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_2467_pow__divides__pow__iff,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_2468_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_2469_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_2470_low__inv,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( 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))),Na))),X)),Na) = X ) ) ).

% low_inv
tff(fact_2471_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2472_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_2473_both__member__options__def,axiom,
    ! [Tb: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Tb),X)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Tb),X)
        | aa_nat_o(vEBT_VEBT_membermima(Tb),X) ) ) ).

% both_member_options_def
tff(fact_2474_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))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_2475_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X4,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_2476_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X4: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X4))
    <=> ? [X4: complex] :
          ( aa_complex_o(dvd_dvd_complex(L),aa_complex_complex(plus_plus_complex(X4),zero_zero_complex))
          & aa_complex_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_2477_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_2478_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_2479_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_2480_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),zero_zero_int))
          & aa_int_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_2481_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2482_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2483_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2484_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2485_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2486_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2487_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X2: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2488_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,X: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Na)),X) ).

% buildup_nothing_in_min_max
tff(fact_2489_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_2490_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(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_2491_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2492_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2493_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_2494_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_2495_semiring__norm_I13_J,axiom,
    ! [M: num,Na: num] : times_times_num(aa_num_num(bit0,M),aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(bit0,times_times_num(M,Na))) ).

% semiring_norm(13)
tff(fact_2496_semiring__norm_I11_J,axiom,
    ! [M: num] : times_times_num(M,one) = M ).

% semiring_norm(11)
tff(fact_2497_semiring__norm_I12_J,axiom,
    ! [Na: num] : times_times_num(one,Na) = Na ).

% semiring_norm(12)
tff(fact_2498_num__double,axiom,
    ! [Na: num] : times_times_num(aa_num_num(bit0,one),Na) = aa_num_num(bit0,Na) ).

% num_double
tff(fact_2499_power__mult__numeral,axiom,
    ! [A: nat,M: num,Na: 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,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,times_times_num(M,Na))) ).

% power_mult_numeral
tff(fact_2500_power__mult__numeral,axiom,
    ! [A: int,M: num,Na: 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,Na)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,times_times_num(M,Na))) ).

% power_mult_numeral
tff(fact_2501_power__mult__numeral,axiom,
    ! [A: real,M: num,Na: num] : aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,times_times_num(M,Na))) ).

% power_mult_numeral
tff(fact_2502_power__mult__numeral,axiom,
    ! [A: complex,M: num,Na: 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,Na)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,times_times_num(M,Na))) ).

% power_mult_numeral
tff(fact_2503_power__mult__numeral,axiom,
    ! [A: rat,M: num,Na: num] : aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Na)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,times_times_num(M,Na))) ).

% power_mult_numeral
tff(fact_2504_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_2505_zdvd__mono,axiom,
    ! [K: int,M: int,Tb: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Tb)
      <=> aa_int_o(aa_int_fun_int_o(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),Tb)) ) ) ).

% zdvd_mono
tff(fact_2506_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_2507_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_2508_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(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),Na))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na) ) ) ).

% zdvd_mult_cancel
tff(fact_2509_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_2510_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_2511_zdvd__reduce,axiom,
    ! [K: int,Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),aa_int_int(aa_int_fun_int_int(times_times_int,K),M)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na) ) ).

% zdvd_reduce
tff(fact_2512_zdvd__period,axiom,
    ! [A: int,D: int,X: int,Tb: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Tb))
      <=> aa_int_o(aa_int_fun_int_o(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),D))),Tb)) ) ) ).

% zdvd_period
tff(fact_2513_imult__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,Na) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( Na = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_2514_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K)),K) ).

% unset_bit_less_eq
tff(fact_2515_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K)) ).

% set_bit_greater_eq
tff(fact_2516_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,times_times_num(K,L))) ).

% div_mult2_numeral_eq
tff(fact_2517_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,times_times_num(K,L))) ).

% div_mult2_numeral_eq
tff(fact_2518_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(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_2519_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,Na))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_2520_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2521_minf_I7_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Tb),X3) ) ).

% minf(7)
tff(fact_2522_minf_I7_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Tb),X3) ) ).

% minf(7)
tff(fact_2523_minf_I7_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z2)
     => ~ aa_num_o(ord_less_num(Tb),X3) ) ).

% minf(7)
tff(fact_2524_minf_I7_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Tb),X3) ) ).

% minf(7)
tff(fact_2525_minf_I7_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3) ) ).

% minf(7)
tff(fact_2526_minf_I5_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Tb) ) ).

% minf(5)
tff(fact_2527_minf_I5_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Tb) ) ).

% minf(5)
tff(fact_2528_minf_I5_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z2)
     => aa_num_o(ord_less_num(X3),Tb) ) ).

% minf(5)
tff(fact_2529_minf_I5_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Tb) ) ).

% minf(5)
tff(fact_2530_minf_I5_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb) ) ).

% minf(5)
tff(fact_2531_minf_I4_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_2532_minf_I4_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_2533_minf_I4_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_2534_minf_I4_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_2535_minf_I4_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_2536_minf_I3_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_2537_minf_I3_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_2538_minf_I3_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_2539_minf_I3_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_2540_minf_I3_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_2541_minf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q5: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q5,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
           => ( ( aa_real_o(P,X3)
                | aa_real_o(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                | aa_real_o(Q5,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_2542_minf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q5: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q5,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
           => ( ( aa_rat_o(P,X3)
                | aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                | aa_rat_o(Q5,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_2543_minf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q5: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(X2),Z3)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(X2),Z3)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q5,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(X3),Z2)
           => ( ( aa_num_o(P,X3)
                | aa_num_o(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                | aa_num_o(Q5,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_2544_minf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q5: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q5,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
           => ( ( aa_nat_o(P,X3)
                | aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                | aa_nat_o(Q5,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_2545_minf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q5: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q5,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                | aa_int_o(Q5,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_2546_minf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q5: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z3)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q5,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
           => ( ( aa_real_o(P,X3)
                & aa_real_o(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                & aa_real_o(Q5,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_2547_minf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q5: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z3)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q5,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
           => ( ( aa_rat_o(P,X3)
                & aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                & aa_rat_o(Q5,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_2548_minf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q5: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(X2),Z3)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(X2),Z3)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q5,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(X3),Z2)
           => ( ( aa_num_o(P,X3)
                & aa_num_o(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                & aa_num_o(Q5,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_2549_minf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q5: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z3)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q5,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
           => ( ( aa_nat_o(P,X3)
                & aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                & aa_nat_o(Q5,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_2550_minf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q5: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q5,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                & aa_int_o(Q5,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_2551_pinf_I7_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Tb),X3) ) ).

% pinf(7)
tff(fact_2552_pinf_I7_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Tb),X3) ) ).

% pinf(7)
tff(fact_2553_pinf_I7_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z2),X3)
     => aa_num_o(ord_less_num(Tb),X3) ) ).

% pinf(7)
tff(fact_2554_pinf_I7_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Tb),X3) ) ).

% pinf(7)
tff(fact_2555_pinf_I7_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3) ) ).

% pinf(7)
tff(fact_2556_pinf_I5_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Tb) ) ).

% pinf(5)
tff(fact_2557_pinf_I5_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Tb) ) ).

% pinf(5)
tff(fact_2558_pinf_I5_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z2),X3)
     => ~ aa_num_o(ord_less_num(X3),Tb) ) ).

% pinf(5)
tff(fact_2559_pinf_I5_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Tb) ) ).

% pinf(5)
tff(fact_2560_pinf_I5_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb) ) ).

% pinf(5)
tff(fact_2561_pinf_I4_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_2562_pinf_I4_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_2563_pinf_I4_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_2564_pinf_I4_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_2565_pinf_I4_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_2566_pinf_I3_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_2567_pinf_I3_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_2568_pinf_I3_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_2569_pinf_I3_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_2570_pinf_I3_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_2571_pinf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q5: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q5,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
           => ( ( aa_real_o(P,X3)
                | aa_real_o(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                | aa_real_o(Q5,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_2572_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q5: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q5,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
           => ( ( aa_rat_o(P,X3)
                | aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                | aa_rat_o(Q5,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_2573_pinf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q5: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(Z3),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(Z3),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q5,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(Z2),X3)
           => ( ( aa_num_o(P,X3)
                | aa_num_o(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                | aa_num_o(Q5,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_2574_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q5: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q5,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
           => ( ( aa_nat_o(P,X3)
                | aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                | aa_nat_o(Q5,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_2575_pinf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q5: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q5,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                | aa_int_o(Q5,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_2576_pinf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q5: fun_real_o] :
      ( ? [Z3: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P2,X2) ) )
     => ( ? [Z3: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q5,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
           => ( ( aa_real_o(P,X3)
                & aa_real_o(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                & aa_real_o(Q5,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_2577_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q5: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P2,X2) ) )
     => ( ? [Z3: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q5,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
           => ( ( aa_rat_o(P,X3)
                & aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                & aa_rat_o(Q5,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_2578_pinf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q5: fun_num_o] :
      ( ? [Z3: num] :
        ! [X2: num] :
          ( aa_num_o(ord_less_num(Z3),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P2,X2) ) )
     => ( ? [Z3: num] :
          ! [X2: num] :
            ( aa_num_o(ord_less_num(Z3),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q5,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(ord_less_num(Z2),X3)
           => ( ( aa_num_o(P,X3)
                & aa_num_o(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                & aa_num_o(Q5,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_2579_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q5: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P2,X2) ) )
     => ( ? [Z3: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q5,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
           => ( ( aa_nat_o(P,X3)
                & aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                & aa_nat_o(Q5,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_2580_pinf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q5: fun_int_o] :
      ( ? [Z3: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P2,X2) ) )
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q5,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                & aa_int_o(Q5,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_2581_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2582_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2583_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2584_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_2585_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_2586_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_2587_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_2588_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_2589_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(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,B3),Q6)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_2590_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R4: int,B3: int,Q6: int,R3: 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)),R4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q6)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(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,B3),Q6)),R3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2591_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R4: int,B3: int,Q6: int,R3: 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)),R4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q6)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(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,B3),Q6)),R3)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2592_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R3: int,Q3: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(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)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2593_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R3: int,Q3: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(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)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R4),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2594_four__x__squared,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_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_2595_split__zdiv,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,I3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,I3) ) ) ) ) ).

% split_zdiv
tff(fact_2596_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R4: 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)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R4),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2597_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R4: 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)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2598_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_2599_pinf_I6_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Tb) ) ).

% pinf(6)
tff(fact_2600_pinf_I6_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z2),X3)
     => ~ aa_num_o(ord_less_eq_num(X3),Tb) ) ).

% pinf(6)
tff(fact_2601_pinf_I6_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Tb) ) ).

% pinf(6)
tff(fact_2602_pinf_I6_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb) ) ).

% pinf(6)
tff(fact_2603_pinf_I6_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Tb) ) ).

% pinf(6)
tff(fact_2604_pinf_I8_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Tb),X3) ) ).

% pinf(8)
tff(fact_2605_pinf_I8_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(Z2),X3)
     => aa_num_o(ord_less_eq_num(Tb),X3) ) ).

% pinf(8)
tff(fact_2606_pinf_I8_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),X3) ) ).

% pinf(8)
tff(fact_2607_pinf_I8_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3) ) ).

% pinf(8)
tff(fact_2608_pinf_I8_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Tb),X3) ) ).

% pinf(8)
tff(fact_2609_minf_I6_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Tb) ) ).

% minf(6)
tff(fact_2610_minf_I6_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z2)
     => aa_num_o(ord_less_eq_num(X3),Tb) ) ).

% minf(6)
tff(fact_2611_minf_I6_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Tb) ) ).

% minf(6)
tff(fact_2612_minf_I6_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb) ) ).

% minf(6)
tff(fact_2613_minf_I6_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Tb) ) ).

% minf(6)
tff(fact_2614_minf_I8_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Tb),X3) ) ).

% minf(8)
tff(fact_2615_minf_I8_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(ord_less_num(X3),Z2)
     => ~ aa_num_o(ord_less_eq_num(Tb),X3) ) ).

% minf(8)
tff(fact_2616_minf_I8_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),X3) ) ).

% minf(8)
tff(fact_2617_minf_I8_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3) ) ).

% minf(8)
tff(fact_2618_minf_I8_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Tb),X3) ) ).

% minf(8)
tff(fact_2619_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] : X != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_2620_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_2621_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B4)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B4),A4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),zero_zero_nat)
       => ( ! [A4: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_2622_imp__le__cong,axiom,
    ! [X: int,X5: int,P: $o,P2: $o] :
      ( ( X = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_2623_conj__le__cong,axiom,
    ! [X: int,X5: int,P: $o,P2: $o] :
      ( ( X = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_2624_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X2: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D3) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_2625_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y2)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_2626_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2627_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_2628_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_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_2629_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_2630_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(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_2631_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(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_2632_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_2633_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(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_2634_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(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_2635_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(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))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2636_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2637_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(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))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2638_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% minf(10)
tff(fact_2639_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% minf(10)
tff(fact_2640_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S)) ) ) ).

% minf(10)
tff(fact_2641_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% minf(10)
tff(fact_2642_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% minf(10)
tff(fact_2643_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% minf(9)
tff(fact_2644_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% minf(9)
tff(fact_2645_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S)) ) ) ).

% minf(9)
tff(fact_2646_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% minf(9)
tff(fact_2647_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% minf(9)
tff(fact_2648_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% pinf(10)
tff(fact_2649_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% pinf(10)
tff(fact_2650_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S)) ) ) ).

% pinf(10)
tff(fact_2651_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% pinf(10)
tff(fact_2652_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% pinf(10)
tff(fact_2653_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)) ) ) ).

% pinf(9)
tff(fact_2654_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% pinf(9)
tff(fact_2655_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),S)) ) ) ).

% pinf(9)
tff(fact_2656_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% pinf(9)
tff(fact_2657_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% pinf(9)
tff(fact_2658_both__member__options__ding,axiom,
    ! [Infoa: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Na: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(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))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Infoa,Dega,TreeLista,Summarya)),X) ) ) ) ).

% both_member_options_ding
tff(fact_2659_low__def,axiom,
    ! [X: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na) = 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))),Na)) ).

% low_def
tff(fact_2660_triangle__def,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_2661_set__decode__0,axiom,
    ! [X: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X) ) ).

% set_decode_0
tff(fact_2662_set__decode__Suc,axiom,
    ! [Na: nat,X: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Na)),aa_nat_set_nat(nat_set_decode,X))
    <=> aa_set_nat_o(member_nat(Na),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_2663_add__scale__eq__noteq,axiom,
    ! [R4: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R4 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R4),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R4),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2664_add__scale__eq__noteq,axiom,
    ! [R4: real,A: real,B: real,C: real,D: real] :
      ( ( R4 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2665_add__scale__eq__noteq,axiom,
    ! [R4: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R4 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(times_times_rat(R4),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(times_times_rat(R4),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2666_add__scale__eq__noteq,axiom,
    ! [R4: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R4 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R4),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R4),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2667_add__scale__eq__noteq,axiom,
    ! [R4: int,A: int,B: int,C: int,D: int] :
      ( ( R4 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R4),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R4),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2668_even__succ__mod__exp,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( 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))),Na)) = 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))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_2669_even__succ__mod__exp,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( 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))),Na)) = 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))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_2670_even__succ__mod__exp,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_2671_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(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,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2672_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(aa_nat_fun_nat_o(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,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2673_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2674_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(pow(K,L)) ).

% power_numeral
tff(fact_2675_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(power_power_real(numeral_numeral_real(K)),aa_num_nat(numeral_numeral_nat,L)) = numeral_numeral_real(pow(K,L)) ).

% power_numeral
tff(fact_2676_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(numeral_numeral_rat(K)),aa_num_nat(numeral_numeral_nat,L)) = numeral_numeral_rat(pow(K,L)) ).

% power_numeral
tff(fact_2677_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,pow(K,L)) ).

% power_numeral
tff(fact_2678_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,pow(K,L)) ).

% power_numeral
tff(fact_2679_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Na: nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,set_complex2(Xs))
         => aa_complex_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Na)) ) ) ).

% inthall
tff(fact_2680_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Na: nat] :
      ( ! [X2: real] :
          ( member_real(X2,set_real2(Xs))
         => aa_real_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Na)) ) ) ).

% inthall
tff(fact_2681_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Na: nat] :
      ( ! [X2: set_nat] :
          ( member_set_nat(X2,set_set_nat2(Xs))
         => aa_set_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_2682_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Na: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% inthall
tff(fact_2683_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Na: nat] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_o_o(P,(X2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% inthall
tff(fact_2684_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Na: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_2685_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Na: nat] :
      ( ! [X2: int] :
          ( member_int(X2,set_int2(Xs))
         => aa_int_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% inthall
tff(fact_2686_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_2687_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mod_trivial
tff(fact_2688_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_2689_real__divide__square__eq,axiom,
    ! [R4: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),R4)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R4) ).

% real_divide_square_eq
tff(fact_2690_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_2691_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_2692_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_2693_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_2694_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_2695_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_2696_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_2697_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_2698_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_2699_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_2700_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_2701_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_2702_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_2703_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_2704_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2705_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_2706_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2707_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_2708_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_2709_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_2710_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).

% add_diff_cancel
tff(fact_2711_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_2712_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_2713_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ).

% diff_add_cancel
tff(fact_2714_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_2715_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_2716_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = minus_minus_rat(A,B) ).

% add_diff_cancel_left
tff(fact_2717_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_2718_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_2719_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_2720_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_2721_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_2722_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_2723_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_2724_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(A,B) ).

% add_diff_cancel_right
tff(fact_2725_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_2726_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_2727_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_2728_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_2729_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_2730_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_2731_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_2732_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_2733_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_2734_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_2735_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_2736_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_2737_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_2738_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_2739_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_2740_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_2741_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_2742_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_2743_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_2744_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_2745_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_2746_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_2747_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_2748_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_2749_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_2750_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_2751_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% diff_Suc_Suc
tff(fact_2752_Suc__diff__diff,axiom,
    ! [M: nat,Na: 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)),Na)),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),Na)),K) ).

% Suc_diff_diff
tff(fact_2753_diff__0__eq__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Na) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_2754_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_2755_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_2756_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_2757_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = M ) ) ).

% mod_less
tff(fact_2758_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_2759_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2760_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2761_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2762_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2763_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2764_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2765_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2766_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2767_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_2768_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_2769_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2770_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2771_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_2772_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_2773_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_2774_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_2775_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2776_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_2777_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_2778_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_2779_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2780_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2781_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_2782_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_2783_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_2784_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_2785_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_2786_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_2787_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_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_2795_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_2796_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2797_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_2798_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_2799_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_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(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_2800_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_2801_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_2802_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_2803_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_2804_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_2805_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_2806_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_2807_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_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(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_2808_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_2809_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_2810_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_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(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_2811_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_2812_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_2813_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_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(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_2814_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_2815_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_2816_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_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(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_2817_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_2818_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2819_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_2820_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_2821_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% zero_less_diff
tff(fact_2822_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% diff_is_0_eq
tff(fact_2823_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2824_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2825_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2826_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2827_diff__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),one_one_nat) = Na ).

% diff_Suc_1
tff(fact_2828_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_2829_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2830_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2831_triangle__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Na)),aa_nat_nat(suc,Na)) ).

% triangle_Suc
tff(fact_2832_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_2833_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_2834_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2835_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2836_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self4
tff(fact_2837_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self3
tff(fact_2838_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self2
tff(fact_2839_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self1
tff(fact_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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_2846_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_2847_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_2848_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_2849_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_2850_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_2851_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: 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)),Na))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2852_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_2853_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(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_2854_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_2855_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_2856_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_2857_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_2858_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_2859_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_2860_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2861_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_2862_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_2863_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(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),Na)) ) ) ).

% even_diff_nat
tff(fact_2864_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_2865_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na),one_one_Code_integer))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2866_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)),one_one_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2867_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))),Na)),one_one_int))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2868_odd__two__times__div__two__nat,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( 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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2869_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_2870_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B))) ).

% dvd_minus_mod
tff(fact_2871_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_2872_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_2873_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_2874_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2875_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2876_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: 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),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2877_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_2878_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(minus_minus_rat(A,C),B) = minus_minus_rat(minus_minus_rat(A,B),C) ).

% diff_right_commute
tff(fact_2879_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_2880_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_2881_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_2882_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_2883_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_2884_mod__diff__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_2885_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B3),C) ) ) ) ).

% mod_diff_cong
tff(fact_2886_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_left_eq
tff(fact_2887_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_2888_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_right_eq
tff(fact_2889_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_2890_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_2891_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% mod_geq
tff(fact_2892_mod__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ).

% mod_if
tff(fact_2893_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_2894_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_2895_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_2896_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_2897_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_2898_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_2899_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_2900_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_2901_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_2902_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_2903_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_2904_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_2905_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,Na),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2906_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na)) ).

% modulo_nat_def
tff(fact_2907_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_2908_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_2909_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_eq
tff(fact_2910_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_2911_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_2912_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A3),B3),C) ) ) ) ).

% mod_mult_cong
tff(fact_2913_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_2914_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_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(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),B)),C) ).

% mod_mult_mult2
tff(fact_2915_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_2916_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_2917_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(modulo_modulo_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_mod_right
tff(fact_2918_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_2919_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_2920_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_2921_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_left_eq
tff(fact_2922_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_2923_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_2924_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_right_eq
tff(fact_2925_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_2926_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_2927_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_2928_mod__add__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_2929_mod__add__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_2930_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B3),C) ) ) ) ).

% mod_add_cong
tff(fact_2931_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_2932_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_2933_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_2934_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_2935_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_2936_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_2937_power__mod,axiom,
    ! [A: nat,B: nat,Na: 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)),Na)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),B) ).

% power_mod
tff(fact_2938_power__mod,axiom,
    ! [A: int,B: int,Na: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),Na)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),Na)),B) ).

% power_mod
tff(fact_2939_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Na),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),B) ).

% power_mod
tff(fact_2940_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2941_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2942_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2943_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2944_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2945_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2946_dvd__mod,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ) ) ).

% dvd_mod
tff(fact_2947_dvd__mod,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(modulo_modulo_int(M),Na)) ) ) ).

% dvd_mod
tff(fact_2948_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,Na: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(M,Na)) ) ) ).

% dvd_mod
tff(fact_2949_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_2950_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),C) = aa_int_int(modulo_modulo_int(A),C) ) ) ).

% mod_mod_cancel
tff(fact_2951_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_2952_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_mono
tff(fact_2953_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% diff_mono
tff(fact_2954_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_2955_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_left_mono
tff(fact_2956_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(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_2957_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(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_2958_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_right_mono
tff(fact_2959_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_2960_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_2961_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2962_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: 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),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2963_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2964_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_2965_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_2966_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2967_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_2968_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Na) ).

% mod_Suc_Suc_eq
tff(fact_2969_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% mod_Suc_eq
tff(fact_2970_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_2971_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_strict_mono
tff(fact_2972_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% diff_strict_mono
tff(fact_2973_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2974_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2975_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: 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),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2976_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(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_2977_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_strict_left_mono
tff(fact_2978_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(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_2979_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(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_2980_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_strict_right_mono
tff(fact_2981_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(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_2982_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_2983_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_2984_left__diff__distrib,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(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(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_2985_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_2986_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_2987_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_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(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_2988_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_2989_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(times_times_rat(B),A),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_2990_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_2991_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_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)) ).

% left_diff_distrib'
tff(fact_2992_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_2993_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_2994_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_2995_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_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(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_2996_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2997_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( minus_minus_rat(A2,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_2998_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),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_2999_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_3000_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat(A,B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_3001_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_3002_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_3003_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = minus_minus_rat(C,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3004_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_3005_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_3006_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) ).

% add_diff_eq
tff(fact_3007_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_3008_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_3009_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).

% diff_diff_eq2
tff(fact_3010_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_3011_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_3012_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).

% diff_add_eq
tff(fact_3013_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_3014_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_3015_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3016_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_3017_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_3018_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = minus_minus_rat(A,B) ) ) ).

% add_implies_diff
tff(fact_3019_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_3020_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_3021_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_3022_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_3023_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_3024_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_3025_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_3026_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),minus_minus_rat(C,D)) ).

% add_diff_add
tff(fact_3027_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: 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),D)) = 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),D)) ).

% add_diff_add
tff(fact_3028_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
     => ? [Q2: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q2) ) ).

% zmod_eq_0D
tff(fact_3029_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
    <=> ? [Q4: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_3030_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_3031_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_3032_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_3033_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(minus_minus_rat(A,B),C) = minus_minus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% diff_divide_distrib
tff(fact_3034_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),minus_8373710615458151222nteger(Y,Z)) ) ) ).

% dvd_diff
tff(fact_3035_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ) ) ).

% dvd_diff
tff(fact_3036_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),minus_minus_rat(Y,Z)) ) ) ).

% dvd_diff
tff(fact_3037_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ) ).

% dvd_diff
tff(fact_3038_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C)) ) ).

% dvd_diff_commute
tff(fact_3039_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_3040_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_3041_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_3042_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M) = zero_zero_nat )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_3043_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_3044_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_3045_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_3046_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(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,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% le_diff_iff
tff(fact_3047_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( 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,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3048_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(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,Na),L)) ) ).

% diff_le_mono
tff(fact_3049_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ).

% diff_le_self
tff(fact_3050_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(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))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_3051_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_3052_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% Nat.diff_cancel
tff(fact_3053_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% diff_cancel2
tff(fact_3054_diff__add__inverse,axiom,
    ! [Na: 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,Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_3055_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_3056_diff__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),K)) ).

% diff_mult_distrib
tff(fact_3057_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ).

% diff_mult_distrib2
tff(fact_3058_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_3059_mod__nat__eqI,axiom,
    ! [R4: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R4),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R4),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R4))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = R4 ) ) ) ) ).

% mod_nat_eqI
tff(fact_3060_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% subset_decode_imp_le
tff(fact_3061_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3062_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(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_3063_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3064_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_3065_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3066_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3067_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_3068_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3069_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_3070_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),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,Na)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_3071_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),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,Na)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_3072_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,Na: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_3073_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Na: 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,Na)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_3074_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Na: 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,Na)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_3075_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_3076_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D3: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D3)) ) ).

% mod_eqE
tff(fact_3077_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D3: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3)) ) ).

% mod_eqE
tff(fact_3078_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 )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3079_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3080_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3081_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3082_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3083_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3084_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 )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3085_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3086_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3087_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_3088_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_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,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_3089_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_3090_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3091_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3092_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3093_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3094_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I3),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I3),nth_VEBT_VEBT(Xs3,I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_3095_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I3),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_o_o(aa_nat_fun_o_o(P,I3),aa_nat_o(nth_o(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_3096_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(nth_nat(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_3097_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I3),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_int_o(aa_nat_fun_int_o(P,I3),aa_nat_int(nth_int(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_3098_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3099_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3100_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3101_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3102_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3103_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(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_3104_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(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_3105_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3106_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3107_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(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_3108_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_3109_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_3110_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_3111_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_3112_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_3113_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_3114_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3115_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_3116_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(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_3117_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3118_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3119_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3120_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3121_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3122_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3123_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3124_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),minus_minus_rat(Na,K)) ) ).

% add_le_imp_le_diff
tff(fact_3125_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_3126_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_3127_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_3128_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Na: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Na,K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3129_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Na: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3130_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Na: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3131_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Na: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3132_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = Na,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))) ).

% mod_Suc
tff(fact_3133_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_3134_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_3135_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_3136_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_3137_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_3138_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_3139_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(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_3140_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3141_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(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_3142_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(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_3143_mod__induct,axiom,
    ! [P: fun_nat_o,Na: nat,P3: nat,M: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P3)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P3)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P3)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_3144_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_3145_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M4),zero_zero_nat)
     => ( ! [M4: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M4),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M4),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_3146_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3147_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3148_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: 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)),D) )
    <=> ( 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) = D ) ) ).

% eq_add_iff1
tff(fact_3149_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3150_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3151_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: 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)),D) )
    <=> ( 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)),D) ) ) ).

% eq_add_iff2
tff(fact_3152_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% square_diff_square_factored
tff(fact_3153_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),minus_minus_rat(X,Y)) ).

% square_diff_square_factored
tff(fact_3154_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_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)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_3155_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_3156_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),Y),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(X),minus_minus_rat(Y,B))),aa_rat_rat(times_times_rat(minus_minus_rat(X,A)),B)) ).

% mult_diff_mult
tff(fact_3157_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_3158_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_3159_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q2: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_3160_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_3161_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_3162_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L)) ) ).

% neg_mod_bound
tff(fact_3163_nat__mod__eq__iff,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3164_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_3165_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_3166_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_3167_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_3168_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(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,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ) ).

% less_diff_iff
tff(fact_3169_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(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_3170_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_3171_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_3172_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_3173_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_3174_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_3175_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3176_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3177_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3178_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = 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)),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_3179_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_3180_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_3181_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_3182_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_3183_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one) = X ).

% pow.simps(1)
tff(fact_3184_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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))),Na)) = 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,Na),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_3185_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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))),Na)) = 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,Na),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_3186_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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))),Na)) = 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,Na),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3187_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3188_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3189_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3190_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3191_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(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_3192_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(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_3193_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),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,Na)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3194_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),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,Na)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3195_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_3196_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_3197_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_3198_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_3199_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_3200_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_3201_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_3202_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Na: 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,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_3203_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Na: 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,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_3204_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_3205_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_3206_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_3207_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_3208_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_3209_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : 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(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_3210_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_3211_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_3212_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_3213_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_3214_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3215_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3216_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3217_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_3218_div__mult__mod__eq,axiom,
    ! [A: int,B: 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(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_3219_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_3220_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_3221_mod__div__decomp,axiom,
    ! [A: int,B: 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_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_3222_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_3223_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3224_cancel__div__mod__rules_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,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3225_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_3226_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3227_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: 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,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3228_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_3229_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3230_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_3231_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_3232_nth__mem,axiom,
    ! [Na: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3451745648224563538omplex(Xs))
     => member_complex(aa_nat_complex(nth_complex(Xs),Na),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_3233_nth__mem,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Na),set_real2(Xs)) ) ).

% nth_mem
tff(fact_3234_nth__mem,axiom,
    ! [Na: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),Na),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3235_nth__mem,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Na),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_3236_nth__mem,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Na),set_o2(Xs)) ) ).

% nth_mem
tff(fact_3237_nth__mem,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Na)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3238_nth__mem,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => member_int(aa_nat_int(nth_int(Xs),Na),set_int2(Xs)) ) ).

% nth_mem
tff(fact_3239_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X2) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% list_ball_nth
tff(fact_3240_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( ! [X2: $o] :
            ( member_o((X2),set_o2(Xs))
           => aa_o_o(P,(X2)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_3241_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => aa_nat_o(P,X2) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_3242_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => ( ! [X2: int] :
            ( member_int(X2,set_int2(Xs))
           => aa_int_o(P,X2) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_3243_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3244_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3245_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3246_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3247_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( member_o((X),set_o2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I3)
          <=> (X) ) ) ) ).

% in_set_conv_nth
tff(fact_3248_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3249_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3250_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,X: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I2)) )
     => ( member_complex(X,set_complex2(Xs))
       => aa_complex_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3251_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,X: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) )
     => ( member_real(X,set_real2(Xs))
       => aa_real_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3252_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,X: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),I2)) )
     => ( member_set_nat(X,set_set_nat2(Xs))
       => aa_set_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3253_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3254_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,X: $o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) )
     => ( member_o((X),set_o2(Xs))
       => aa_o_o(P,(X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3255_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,X: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
       => aa_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3256_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,X: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) )
     => ( member_int(X,set_int2(Xs))
       => aa_int_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3257_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_3258_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => aa_o_o(P,(X4)) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_3259_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_3260_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => aa_int_o(P,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_3261_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3262_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(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)),D))
    <=> aa_int_o(aa_int_fun_int_o(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)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3263_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(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)),D))
    <=> aa_real_o(aa_real_fun_real_o(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)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3264_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3265_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(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)),D))
    <=> aa_int_o(aa_int_fun_int_o(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)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3266_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(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)),D))
    <=> aa_real_o(aa_real_fun_real_o(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)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3267_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(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)),D))
    <=> aa_real_o(aa_real_fun_real_o(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)),D)) ) ).

% less_add_iff2
tff(fact_3268_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3269_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(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)),D))
    <=> aa_int_o(aa_int_fun_int_o(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)),D)) ) ).

% less_add_iff2
tff(fact_3270_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(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)),D))
    <=> aa_real_o(aa_real_fun_real_o(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)),D) ) ).

% less_add_iff1
tff(fact_3271_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3272_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(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)),D))
    <=> aa_int_o(aa_int_fun_int_o(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)),D) ) ).

% less_add_iff1
tff(fact_3273_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,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_3274_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,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_3275_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      minus_minus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3276_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_3277_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_3278_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3279_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_3280_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_3281_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3282_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_3283_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_3284_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(minus_minus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3285_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3286_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).

% square_diff_one_factored
tff(fact_3287_square__diff__one__factored,axiom,
    ! [X: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),minus_minus_rat(X,one_one_rat)) ).

% square_diff_one_factored
tff(fact_3288_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_3289_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_3290_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,Tb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X3: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,Tb))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Tb)) ) ) ).

% inf_period(3)
tff(fact_3291_inf__period_I3_J,axiom,
    ! [D: real,D4: real,Tb: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X3: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Tb))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Tb)) ) ) ).

% inf_period(3)
tff(fact_3292_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,Tb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X3: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Tb))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K4),D4))),Tb)) ) ) ).

% inf_period(3)
tff(fact_3293_inf__period_I3_J,axiom,
    ! [D: int,D4: int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Tb)) ) ) ).

% inf_period(3)
tff(fact_3294_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,Tb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X3: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,Tb))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Tb)) ) ) ).

% inf_period(4)
tff(fact_3295_inf__period_I4_J,axiom,
    ! [D: real,D4: real,Tb: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X3: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Tb))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Tb)) ) ) ).

% inf_period(4)
tff(fact_3296_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,Tb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X3: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Tb))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K4),D4))),Tb)) ) ) ).

% inf_period(4)
tff(fact_3297_inf__period_I4_J,axiom,
    ! [D: int,D4: int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Tb)) ) ) ).

% inf_period(4)
tff(fact_3298_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3299_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Na: 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,Na),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_3300_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Na: 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,Na),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ~ ! [S3: nat] : Na != 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_3301_nat__mod__eq__lemma,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ? [Q2: 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,Na),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3302_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3303_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_3304_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_3305_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_3306_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% neg_mod_conj
tff(fact_3307_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ).

% mod_mult2_eq
tff(fact_3308_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_3309_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => aa_nat_o(P,D5) ) ) ) ).

% nat_diff_split
tff(fact_3310_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ aa_nat_o(P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_3311_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_3312_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3313_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(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_3314_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(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_3315_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na) )
      <=> ( 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) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_3316_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na) )
      <=> ( 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)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3317_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(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)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_3318_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(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)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_3319_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na)) = 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)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_3320_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na)) = 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)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_3321_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(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,Na),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_3322_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3323_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(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))
        <=> aa_real_o(aa_real_fun_real_o(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_3324_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(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))
        <=> aa_real_o(aa_real_fun_real_o(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_3325_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3326_even__even__mod__4__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_3327_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_3328_power2__commute,axiom,
    ! [X: real,Y: real] : aa_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(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_3329_power2__commute,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3330_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_3331_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Na != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I3)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_3332_power__diff,axiom,
    ! [A: complex,Na: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Na)) ) ) ) ).

% power_diff
tff(fact_3333_power__diff,axiom,
    ! [A: real,Na: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),Na)) ) ) ) ).

% power_diff
tff(fact_3334_power__diff,axiom,
    ! [A: rat,Na: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_diff
tff(fact_3335_power__diff,axiom,
    ! [A: nat,Na: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),Na)) = 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),Na)) ) ) ) ).

% power_diff
tff(fact_3336_power__diff,axiom,
    ! [A: int,Na: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_diff
tff(fact_3337_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(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_3338_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R4: 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)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_int(modulo_modulo_int(A),B) = R4 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3339_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R4: 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)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R4),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
         => ( aa_int_int(modulo_modulo_int(A),B) = R4 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3340_split__zmod,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_3341_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_3342_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3343_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3344_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3345_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        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),Na)),Na)) ) ).

% div_if
tff(fact_3346_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = 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),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_3347_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = $ite(M = zero_zero_nat,Na,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)),Na))) ).

% add_eq_if
tff(fact_3348_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(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)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_3349_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(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)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_3350_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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)),Na))) ).

% mult_eq_if
tff(fact_3351_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R4: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R4),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R4),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_3352_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3353_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3354_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3355_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_3356_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_3357_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3358_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(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_3359_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(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_3360_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3361_scaling__mono,axiom,
    ! [U: rat,V: rat,R4: rat,S: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R4)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R4),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),divide_divide_rat(aa_rat_rat(times_times_rat(R4),minus_minus_rat(V,U)),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3362_scaling__mono,axiom,
    ! [U: real,V: real,R4: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R4),S)
         => aa_real_o(aa_real_fun_real_o(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,R4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3363_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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),Na))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3364_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: 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))),Na) != 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,Na),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3365_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != 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,Na),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3366_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( 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),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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),Na)),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,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3367_power__diff__power__eq,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3368_power__eq__if,axiom,
    ! [P3: complex,M: nat] :
      aa_nat_complex(power_power_complex(P3),M) = $ite(M = zero_zero_nat,one_one_complex,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_3369_power__eq__if,axiom,
    ! [P3: real,M: nat] :
      aa_nat_real(power_power_real(P3),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_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_3370_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,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_3371_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,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_3372_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      aa_nat_int(power_power_int(P3),M) = $ite(M = zero_zero_nat,one_one_int,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_3373_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(modulo_modulo_int(Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_3374_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(modulo_modulo_int(Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_3375_power__minus__mult,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% power_minus_mult
tff(fact_3376_power__minus__mult,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_real(power_power_real(A),Na) ) ) ).

% power_minus_mult
tff(fact_3377_power__minus__mult,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% power_minus_mult
tff(fact_3378_power__minus__mult,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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,Na),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) ) ) ).

% power_minus_mult
tff(fact_3379_power__minus__mult,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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,Na),one_one_nat))),A) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_minus_mult
tff(fact_3380_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),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),Na))) ) ).

% diff_le_diff_pow
tff(fact_3381_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = 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),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_3382_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3383_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(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_3384_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3385_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_3386_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,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_3387_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_3388_mod2__eq__if,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))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_3389_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_3390_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_3391_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(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 ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(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_3392_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(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 ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(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_3393_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3394_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Na: 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))),Na))),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,Na),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_3395_div__exp__mod__exp__eq,axiom,
    ! [A: int,Na: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_3396_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Na: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)),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,Na),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) ).

% div_exp_mod_exp_eq
tff(fact_3397_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3398_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3399_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3400_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_3401_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_3402_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_3403_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_3404_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_3405_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3406_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3407_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3408_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_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,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3409_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_3410_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_3411_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_3412_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_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),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_3413_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_3414_power2__diff,axiom,
    ! [X: real,Y: real] : aa_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3415_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_rat(aa_rat_rat(aa_rat_fun_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(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3416_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_3417_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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_3418_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3419_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3420_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(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_3421_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(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_3422_unset__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Na),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(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3423_unset__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Na)),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(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_3424_unset__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Na)),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,Na),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_3425_set__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Na),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(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3426_set__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Na)),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(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_3427_set__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Na)),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,Na),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_3428_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_3429_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_3430_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_3431_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))),Na)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_3432_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_3433_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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))),Na)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_3434_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_3435_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(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_3436_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3437_in__children__def,axiom,
    ! [Na: nat,TreeLista: list_VEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Na,TreeLista),X)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)) ) ).

% in_children_def
tff(fact_3438_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Na)) ) ) ).

% verit_le_mono_div
tff(fact_3439_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),
                $ite(aa_int_int(modulo_modulo_int(B2),Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_3440_div__mod__decomp,axiom,
    ! [A2: nat,Na: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na)) ).

% div_mod_decomp
tff(fact_3441_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_3442_flip__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Na),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(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3443_flip__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Na)),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(Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_3444_flip__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Na)),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,Na),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_3445_div__mod__decomp__int,axiom,
    ! [A2: int,Na: int] : A2 = 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(divide_divide_int,A2),Na)),Na)),aa_int_int(modulo_modulo_int(A2),Na)) ).

% div_mod_decomp_int
tff(fact_3446_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_3447_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_3448_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_3449_idiff__0,axiom,
    ! [Na: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Na) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_3450_idiff__0__right,axiom,
    ! [Na: extended_enat] : minus_3235023915231533773d_enat(Na,zero_z5237406670263579293d_enat) = Na ).

% idiff_0_right
tff(fact_3451_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3452_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_3453_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_3454_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_3455_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_3456_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_3457_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_3458_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_3459_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_3460_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),Na))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_3461_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_3462_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_3463_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_3464_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_3465_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
     => ( plus_p3455044024723400733d_enat(X,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_3466_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_3467_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_3468_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_3469_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(ord_less_eq_num(A),B)
      | ~ aa_num_o(ord_less_eq_num(B),A) ) ).

% verit_la_disequality
tff(fact_3470_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_3471_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_3472_verit__la__disequality,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
      | ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      | ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% verit_la_disequality
tff(fact_3473_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_3474_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3475_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(ord_less_eq_num(A),A) ).

% verit_comp_simplify1(2)
tff(fact_3476_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3477_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3478_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3479_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(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_3480_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3481_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3482_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(ord_less_num(A),A) ).

% verit_comp_simplify1(1)
tff(fact_3483_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3484_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3485_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(P1,X2)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
       => ( ? [Z3: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P1,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_3486_plusinfinity,axiom,
    ! [D: int,P2: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(P2,X2)
          <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
       => ( ? [Z3: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P2,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P2,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_3487_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),X)
    <=> $ite(
          X = zero_zero_nat,
          (A),
          $ite(X = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_3488_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_3489_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
      | ( A = X )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A) ) ).

% verit_la_generic
tff(fact_3490_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3491_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(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_3492_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_3493_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B3),A3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_3494_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A3: num] :
      ( ~ aa_num_o(ord_less_eq_num(B3),A3)
    <=> aa_num_o(ord_less_num(A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_3495_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B3),A3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_3496_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),A3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_3497_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),A3)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_3498_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_3499_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_3500_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_3501_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_3502_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_3503_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_3504_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(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_3505_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3506_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(M),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(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_3507_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(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_3508_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => ( ( aa_int_int(modulo_modulo_int(A2),Na) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3509_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(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)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_3510_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(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)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_3511_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Na),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(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_3512_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A4: real,B4: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A4),B4)
           => ( aa_real_o(aa_real_fun_real_o(P,B4),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),B4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A4),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
               => ? [D6: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D6)
                    & ! [A4: real,B4: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B4)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A4)),D6) )
                       => aa_real_o(aa_real_fun_real_o(P,A4),B4) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_3513_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))))) ).

% flip_bit_0
tff(fact_3514_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3515_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(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))))) ).

% flip_bit_0
tff(fact_3516_signed__take__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Na),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(Na,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3517_signed__take__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),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(Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3518_take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Na,A) = $ite(Na = zero_zero_nat,zero_z3403309356797280102nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_3519_take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),A) = $ite(Na = zero_zero_nat,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,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(aa_int_fun_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_3520_take__bit__rec,axiom,
    ! [Na: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = $ite(Na = zero_zero_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_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_3521_take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_3522_take__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_3523_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3524_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3525_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3526_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3527_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),aa_o_Code_integer(zero_n356916108424825756nteger,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_3528_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_3529_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_3530_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_3531_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_3532_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_3533_of__bool__eq_I1_J,axiom,
    aa_o_Code_integer(zero_n356916108424825756nteger,$false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_3534_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3535_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3536_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3537_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3538_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = zero_zero_int )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3539_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_Code_integer(zero_n356916108424825756nteger,(P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_3540_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3541_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3542_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3543_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3544_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),aa_o_Code_integer(zero_n356916108424825756nteger,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_3545_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3546_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3547_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3548_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3549_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3550_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_Code_integer(zero_n356916108424825756nteger,(P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3551_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_3552_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_3553_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_3554_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_3555_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_3556_of__bool__eq_I2_J,axiom,
    aa_o_Code_integer(zero_n356916108424825756nteger,$true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_3557_signed__take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_ri631733984087533419it_int(Na),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_3558_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_3559_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_concat_bit(Na,K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(Na),K) ).

% concat_bit_of_zero_2
tff(fact_3560_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_Suc_0
tff(fact_3561_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_3562_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_3563_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_3564_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_3565_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_3566_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_3567_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_3568_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_3569_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_3570_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_3571_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_3572_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_3573_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_3574_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_3575_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_3576_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_3577_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex((P))) ).

% of_bool_not_iff
tff(fact_3578_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_3579_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_3580_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_3581_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_Code_integer(zero_n356916108424825756nteger,~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,aa_o_Code_integer(zero_n356916108424825756nteger,(P))) ).

% of_bool_not_iff
tff(fact_3582_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_o_nat(zero_n2687167440665602831ol_nat,Na != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_3583_signed__take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_3584_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_3585_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Na,K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_3586_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Na,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_3587_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),one_one_int) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_3588_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = zero_zero_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_3589_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),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(Na),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_3590_odd__of__bool__self,axiom,
    ! [P3: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> (P3) ) ).

% odd_of_bool_self
tff(fact_3591_odd__of__bool__self,axiom,
    ! [P3: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> (P3) ) ).

% odd_of_bool_self
tff(fact_3592_odd__of__bool__self,axiom,
    ! [P3: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(P3)))
    <=> (P3) ) ).

% odd_of_bool_self
tff(fact_3593_take__bit__of__1,axiom,
    ! [Na: nat] : bit_se1745604003318907178nteger(Na,one_one_Code_integer) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_3594_take__bit__of__1,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_3595_take__bit__of__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_3596_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_3597_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_3598_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_3599_even__take__bit__eq,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Na,A))
    <=> ( ( Na = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_3600_even__take__bit__eq,axiom,
    ! [Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(Na),A))
    <=> ( ( Na = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_3601_even__take__bit__eq,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),A))
    <=> ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_3602_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_3603_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_3604_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_3605_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_3606_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_3607_one__div__2__pow__eq,axiom,
    ! [Na: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_3608_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_3609_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_3610_bits__1__div__exp,axiom,
    ! [Na: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_3611_take__bit__of__exp,axiom,
    ! [M: nat,Na: nat] : bit_se1745604003318907178nteger(M,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_3612_take__bit__of__exp,axiom,
    ! [M: nat,Na: nat] : aa_int_int(bit_se2923211474154528505it_int(M),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_3613_take__bit__of__exp,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_3614_take__bit__of__2,axiom,
    ! [Na: nat] : bit_se1745604003318907178nteger(Na,numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_3615_take__bit__of__2,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_3616_take__bit__of__2,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_3617_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_3618_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_3619_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_3620_take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_int_int(bit_se2923211474154528505it_int(Na),L))) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_3621_signed__take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_int_int(bit_ri631733984087533419it_int(Na),L))) = aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_3622_signed__take__bit__take__bit,axiom,
    ! [M: nat,Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(M),aa_int_int(bit_se2923211474154528505it_int(Na),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),bit_se2923211474154528505it_int(Na),bit_ri631733984087533419it_int(M)),A) ).

% signed_take_bit_take_bit
tff(fact_3623_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Na: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Na),A) = aa_int_int(bit_ri631733984087533419it_int(Na),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_3624_of__bool__eq__iff,axiom,
    ! [P3: $o,Q3: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P3)) = aa_o_nat(zero_n2687167440665602831ol_nat,(Q3)) )
    <=> ( (P3)
      <=> (Q3) ) ) ).

% of_bool_eq_iff
tff(fact_3625_of__bool__eq__iff,axiom,
    ! [P3: $o,Q3: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P3)) = aa_o_int(zero_n2684676970156552555ol_int,(Q3)) )
    <=> ( (P3)
      <=> (Q3) ) ) ).

% of_bool_eq_iff
tff(fact_3626_of__bool__eq__iff,axiom,
    ! [P3: $o,Q3: $o] :
      ( ( aa_o_Code_integer(zero_n356916108424825756nteger,(P3)) = aa_o_Code_integer(zero_n356916108424825756nteger,(Q3)) )
    <=> ( (P3)
      <=> (Q3) ) ) ).

% of_bool_eq_iff
tff(fact_3627_concat__bit__eq__iff,axiom,
    ! [Na: nat,K: int,L: int,R4: int,S: int] :
      ( ( aa_int_int(bit_concat_bit(Na,K),L) = aa_int_int(bit_concat_bit(Na,R4),S) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = aa_int_int(bit_se2923211474154528505it_int(Na),R4) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_3628_concat__bit__take__bit__eq,axiom,
    ! [Na: nat,B: int] : bit_concat_bit(Na,aa_int_int(bit_se2923211474154528505it_int(Na),B)) = bit_concat_bit(Na,B) ).

% concat_bit_take_bit_eq
tff(fact_3629_take__bit__add,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Na),A)),aa_int_int(bit_se2923211474154528505it_int(Na),B))) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_3630_take__bit__add,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_3631_take__bit__tightened,axiom,
    ! [Na: nat,A: int,B: int,M: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = aa_int_int(bit_se2923211474154528505it_int(Na),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_int(bit_se2923211474154528505it_int(M),A) = aa_int_int(bit_se2923211474154528505it_int(M),B) ) ) ) ).

% take_bit_tightened
tff(fact_3632_take__bit__tightened,axiom,
    ! [Na: nat,A: nat,B: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( 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_3633_take__bit__signed__take__bit,axiom,
    ! [M: nat,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri631733984087533419it_int(Na),A)) = aa_int_int(bit_se2923211474154528505it_int(M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_3634_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_3635_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(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,Na),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_3636_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_3637_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q) )) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_3638_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_3639_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_3640_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_Code_integer(zero_n356916108424825756nteger,
        ( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),aa_o_Code_integer(zero_n356916108424825756nteger,(Q))) ).

% of_bool_conj
tff(fact_3641_take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_int_int(bit_se2923211474154528505it_int(Na),L))) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_3642_signed__take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_int_int(bit_ri631733984087533419it_int(Na),L))) = aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_3643_signed__take__bit__add,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_int_int(bit_ri631733984087533419it_int(Na),L))) = aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_3644_concat__bit__assoc,axiom,
    ! [Na: nat,K: int,M: nat,L: int,R4: int] : aa_int_int(bit_concat_bit(Na,K),aa_int_int(bit_concat_bit(M,L),R4)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na),aa_int_int(bit_concat_bit(Na,K),L)),R4) ).

% concat_bit_assoc
tff(fact_3645_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).

% zero_less_eq_of_bool
tff(fact_3646_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).

% zero_less_eq_of_bool
tff(fact_3647_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_3648_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% zero_less_eq_of_bool
tff(fact_3649_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(P))) ).

% zero_less_eq_of_bool
tff(fact_3650_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_3651_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_3652_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_3653_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_3654_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_3655_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P3)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_3656_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P3)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_3657_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P3)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_3658_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P3)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_3659_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P3)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_3660_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P3: $o] :
      ( aa_Code_integer_o(P,aa_o_Code_integer(zero_n356916108424825756nteger,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P3)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_3661_split__of__bool,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ( ( (P3)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P3)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_3662_split__of__bool,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P3)))
    <=> ( ( (P3)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P3)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_3663_split__of__bool,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P3)))
    <=> ( ( (P3)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P3)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_3664_split__of__bool,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ( ( (P3)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P3)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_3665_split__of__bool,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ( ( (P3)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P3)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_3666_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P3: $o] :
      ( aa_Code_integer_o(P,aa_o_Code_integer(zero_n356916108424825756nteger,(P3)))
    <=> ( ( (P3)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P3)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_3667_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n1201886186963655149omplex((P3)) = $ite((P3),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_3668_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n3304061248610475627l_real((P3)) = $ite((P3),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_3669_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n2052037380579107095ol_rat((P3)) = $ite((P3),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_3670_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P3)) = $ite((P3),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_3671_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P3)) = $ite((P3),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_3672_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_Code_integer(zero_n356916108424825756nteger,(P3)) = $ite((P3),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_3673_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_3674_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_3675_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% take_bit_nonnegative
tff(fact_3676_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Na),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(Na),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_3677_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_3678_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_3679_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_int_int(bit_se2923211474154528505it_int(Na),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_unset_bit_eq
tff(fact_3680_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_unset_bit_eq
tff(fact_3681_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_int_int(bit_se2923211474154528505it_int(Na),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_set_bit_eq
tff(fact_3682_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_set_bit_eq
tff(fact_3683_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_se2159334234014336723it_int(M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_int_int(bit_se2923211474154528505it_int(Na),A),aa_int_int(bit_se2159334234014336723it_int(M),aa_int_int(bit_se2923211474154528505it_int(Na),A))) ).

% take_bit_flip_bit_eq
tff(fact_3684_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A))) ).

% take_bit_flip_bit_eq
tff(fact_3685_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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))),Na)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_3686_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_3687_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),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,Na),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_3688_take__bit__eq__mod,axiom,
    ! [Na: nat,A: code_integer] : bit_se1745604003318907178nteger(Na,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) ).

% take_bit_eq_mod
tff(fact_3689_take__bit__eq__mod,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Na),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))),Na)) ).

% take_bit_eq_mod
tff(fact_3690_take__bit__eq__mod,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),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))),Na)) ).

% take_bit_eq_mod
tff(fact_3691_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_3692_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_3693_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(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))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_3694_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),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))),Na)) ).

% take_bit_nat_def
tff(fact_3695_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(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))) ).

% of_bool_odd_eq_mod_2
tff(fact_3696_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(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))) ).

% of_bool_odd_eq_mod_2
tff(fact_3697_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_3698_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_less_exp
tff(fact_3699_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Na),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))),Na)) ).

% take_bit_int_def
tff(fact_3700_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_3701_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(M,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3702_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(M),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3703_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Na,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_3704_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_3705_take__bit__eq__0__iff,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(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))),Na)),A) ) ).

% take_bit_eq_0_iff
tff(fact_3706_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(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))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_3707_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
         => aa_nat_o(P,A4) )
     => ( ! [A4: nat,B4: $o] :
            ( aa_nat_o(P,A4)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B4))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B4))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_3708_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
         => aa_int_o(P,A4) )
     => ( ! [A4: int,B4: $o] :
            ( aa_int_o(P,A4)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_3709_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
         => aa_Code_integer_o(P,A4) )
     => ( ! [A4: code_integer,B4: $o] :
            ( aa_Code_integer_o(P,A4)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_3710_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_3711_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_3712_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),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))),Na)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Na),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(Na),K)) ) ) ).

% take_bit_incr_eq
tff(fact_3713_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),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,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_3714_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3715_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3716_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Na),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))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_3717_exp__mod__exp,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,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))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_3718_exp__mod__exp,axiom,
    ! [M: nat,Na: nat] : aa_int_int(modulo_modulo_int(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))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_3719_exp__mod__exp,axiom,
    ! [M: nat,Na: nat] : modulo364778990260209775nteger(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))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_3720_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_3721_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_3722_take__bit__Suc,axiom,
    ! [Na: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Na),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(Na,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_3723_take__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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(Na),aa_int_int(aa_int_fun_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_3724_take__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),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,Na),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_3725_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(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))),Na))),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_3726_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Na,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_3727_stable__imp__take__bit__eq,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_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))),Na)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_3728_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Na: 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(bit_se2925701944663578781it_nat,Na),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),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))),Na)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_3729_exp__div__exp__eq,axiom,
    ! [M: nat,Na: 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))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_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 )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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(minus_minus_nat,M),Na))) ).

% exp_div_exp_eq
tff(fact_3730_exp__div__exp__eq,axiom,
    ! [M: nat,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,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))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ))),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,M),Na))) ).

% exp_div_exp_eq
tff(fact_3731_exp__div__exp__eq,axiom,
    ! [M: nat,Na: nat] :
      divide6298287555418463151nteger(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))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          aa_o_Code_integer(zero_n356916108424825756nteger,
            ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ).

% exp_div_exp_eq
tff(fact_3732_signed__take__bit__rec,axiom,
    ! [Na: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Na,A) = $ite(Na = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),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,Na),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_3733_signed__take__bit__rec,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Na),A) = $ite(Na = zero_zero_nat,aa_int_int(uminus_uminus_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(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,Na),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_3734_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X3),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                   => ( ( 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))),M4) )
                     => ( ( M4 = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X3),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                     => ( ( 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))),M4) )
                       => ( ( M4 = aa_nat_nat(suc,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat,Mi: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X3),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                       => ( ( 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))),M4) )
                         => ( ( M4 = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(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))),M4))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma2)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(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))),Deg))
                                     => ~ ( ( Mi != Ma2 )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(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))),M4))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat,Mi: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X3),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M4)
                         => ( ( 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))),M4) )
                           => ( ( M4 = aa_nat_nat(suc,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(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))),M4))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma2)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(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))),Deg))
                                       => ~ ( ( Mi != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(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))),M4))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3735_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( 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(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma3)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(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))),N4))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(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))
            & ( ( Mi2 != Ma3 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(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))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma3)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( 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(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(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))),aa_nat_nat(suc,N4)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(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))
            & ( ( Mi2 != Ma3 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(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))),aa_nat_nat(suc,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3736_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mi3: nat,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Dega,TreeLista,Summarya)),X) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3737_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_3738_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(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,Na),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_3739_Bernoulli__inequality__even,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(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,Na)),X))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_3740_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_3741_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_3742_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_3743_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_3744_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3745_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_3746_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_3747_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_3748_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_3749_add_Oinverse__inverse,axiom,
    ! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).

% add.inverse_inverse
tff(fact_3750_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_3751_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_3752_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_3753_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_3754_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_3755_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_3756_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_3757_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_3758_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_3759_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_3760_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_3761_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_3762_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_3763_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_3764_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_3765_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_3766_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_3767_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_3768_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = uminus_uminus_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_3769_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_3770_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_3771_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_3772_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_3773_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_3774_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_3775_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_3776_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_3777_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_3778_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3779_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_3780_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_3781_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_3782_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_3783_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_3784_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_3785_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_3786_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_3787_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_3788_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_3789_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: 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,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_3790_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_3791_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_3792_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(M)) = uminus_uminus_rat(numeral_numeral_rat(Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_3793_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_3794_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_3795_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_3796_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_3797_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_3798_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_3799_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_3800_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_3801_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_3802_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_3803_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_3804_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_3805_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_3806_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_3807_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_3808_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_3809_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_3810_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_3811_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_3812_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),uminus_uminus_rat(B)) ).

% minus_add_distrib
tff(fact_3813_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_3814_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_3815_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_3816_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_3817_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_3818_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_3819_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_3820_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_3821_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_3822_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),B)) = B ).

% add_minus_cancel
tff(fact_3823_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_3824_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_3825_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_3826_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_3827_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).

% minus_diff_eq
tff(fact_3828_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% div_minus_minus
tff(fact_3829_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_3830_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y) ) ).

% minus_dvd_iff
tff(fact_3831_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y) ) ).

% minus_dvd_iff
tff(fact_3832_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)
    <=> aa_complex_o(dvd_dvd_complex(X),Y) ) ).

% minus_dvd_iff
tff(fact_3833_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y) ) ).

% minus_dvd_iff
tff(fact_3834_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,uminus_uminus_rat(X)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y) ) ).

% minus_dvd_iff
tff(fact_3835_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y) ) ).

% dvd_minus_iff
tff(fact_3836_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y) ) ).

% dvd_minus_iff
tff(fact_3837_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(X),aa_complex_complex(uminus1482373934393186551omplex,Y))
    <=> aa_complex_o(dvd_dvd_complex(X),Y) ) ).

% dvd_minus_iff
tff(fact_3838_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y) ) ).

% dvd_minus_iff
tff(fact_3839_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),uminus_uminus_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),Y) ) ).

% dvd_minus_iff
tff(fact_3840_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : aa_num_num(bit0,M) != aa_num_num(bit1,Na) ).

% semiring_norm(88)
tff(fact_3841_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : aa_num_num(bit1,M) != aa_num_num(bit0,Na) ).

% semiring_norm(89)
tff(fact_3842_semiring__norm_I84_J,axiom,
    ! [Na: num] : one != aa_num_num(bit1,Na) ).

% semiring_norm(84)
tff(fact_3843_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_3844_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_3845_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_3846_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_3847_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_3848_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_3849_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% real_sqrt_less_iff
tff(fact_3850_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% real_sqrt_le_iff
tff(fact_3851_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y4)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_3852_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y4: num] : X != some_num(Y4)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_3853_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_3854_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y4: num] : X = some_num(Y4) ) ).

% not_None_eq
tff(fact_3855_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_3856_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_3857_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(73)
tff(fact_3858_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(80)
tff(fact_3859_mi__ma__2__deg,axiom,
    ! [Mi3: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Dega,TreeLista,Summarya)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(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))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_3860_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_3861_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_3862_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_3863_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_3864_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_3865_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_3866_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_3867_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_3868_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_3869_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_3870_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_3871_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_3872_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_3873_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_3874_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_3875_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_3876_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_3877_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_3878_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_3879_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_3880_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_3881_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_3882_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_3883_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_3884_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_3885_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_3886_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_3887_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_3888_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_3889_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_3890_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_3891_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_3892_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_3893_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_3894_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_3895_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_3896_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_3897_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_3898_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_3899_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_3900_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_3901_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_3902_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_3903_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_3904_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_3905_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_3906_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : minus_minus_rat(zero_zero_rat,B) = uminus_uminus_rat(B) ).

% verit_minus_simplify(3)
tff(fact_3907_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_3908_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_3909_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_3910_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_3911_diff__0,axiom,
    ! [A: rat] : minus_minus_rat(zero_zero_rat,A) = uminus_uminus_rat(A) ).

% diff_0
tff(fact_3912_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_3913_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: 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,Na))) = 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,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_3914_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_3915_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_3916_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Na))) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Na))) ).

% add_neg_numeral_simps(3)
tff(fact_3917_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_3918_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_3919_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_3920_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_3921_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_3922_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_3923_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_3924_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_3925_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_3926_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_3927_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_3928_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_3929_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_3930_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_3931_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),B) = minus_minus_rat(B,A) ).

% uminus_add_conv_diff
tff(fact_3932_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_3933_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_3934_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_3935_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_3936_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,uminus_uminus_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_3937_divide__minus1,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_3938_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_3939_divide__minus1,axiom,
    ! [X: rat] : divide_divide_rat(X,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(X) ).

% divide_minus1
tff(fact_3940_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_3941_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_3942_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_3943_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_3944_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_3945_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_3946_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_3947_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_3948_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_3949_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_3950_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_3951_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_3952_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_3953_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_3954_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_3955_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_3956_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_3957_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_3958_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_3959_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_3960_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_3961_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_3962_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_3963_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_3964_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_3965_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Na)) = numeral_numeral_real(Na) ).

% of_nat_numeral
tff(fact_3966_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% of_nat_numeral
tff(fact_3967_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,Na) ).

% of_nat_numeral
tff(fact_3968_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Na)) = numera6690914467698888265omplex(Na) ).

% of_nat_numeral
tff(fact_3969_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Na)) = numeral_numeral_rat(Na) ).

% of_nat_numeral
tff(fact_3970_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_3971_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_3972_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_add
tff(fact_3973_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_add
tff(fact_3974_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_add
tff(fact_3975_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% of_nat_add
tff(fact_3976_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_add
tff(fact_3977_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_mult
tff(fact_3978_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mult
tff(fact_3979_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mult
tff(fact_3980_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% of_nat_mult
tff(fact_3981_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_mult
tff(fact_3982_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3983_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3984_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3985_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_3986_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3987_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3988_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3989_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3990_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3991_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3992_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3993_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3994_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3995_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Na) = one_one_complex )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3996_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3997_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X) = aa_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_3998_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_3999_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_4000_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_4001_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_4002_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_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_4003_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_4004_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_4005_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_4006_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_4007_of__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,M)),Na) ).

% of_nat_power
tff(fact_4008_of__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),Na) ).

% of_nat_power
tff(fact_4009_of__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),Na) ).

% of_nat_power
tff(fact_4010_of__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),Na) ).

% of_nat_power
tff(fact_4011_of__nat__power,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Na)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),Na) ).

% of_nat_power
tff(fact_4012_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_4013_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_4014_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_4015_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_4016_signed__take__bit__of__minus__1,axiom,
    ! [Na: nat] : bit_ri6519982836138164636nteger(Na,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4017_signed__take__bit__of__minus__1,axiom,
    ! [Na: nat] : aa_int_int(bit_ri631733984087533419it_int(Na),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_4018_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_4019_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_4020_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_4021_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_4022_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = aa_num_num(bit1,plus_plus_num(M,Na)) ).

% semiring_norm(9)
tff(fact_4023_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = aa_num_num(bit1,plus_plus_num(M,Na)) ).

% semiring_norm(7)
tff(fact_4024_semiring__norm_I15_J,axiom,
    ! [M: num,Na: num] : times_times_num(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = aa_num_num(bit0,times_times_num(aa_num_num(bit1,M),Na)) ).

% semiring_norm(15)
tff(fact_4025_semiring__norm_I14_J,axiom,
    ! [M: num,Na: num] : times_times_num(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = aa_num_num(bit0,times_times_num(M,aa_num_num(bit1,Na))) ).

% semiring_norm(14)
tff(fact_4026_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(72)
tff(fact_4027_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(81)
tff(fact_4028_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_4029_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(ord_less_num(one),aa_num_num(bit1,Na)) ).

% semiring_norm(77)
tff(fact_4030_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K))) ).

% dbl_simps(1)
tff(fact_4031_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_4032_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_4033_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_4034_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_numeral_dbl_rat(numeral_numeral_rat(K))) ).

% dbl_simps(1)
tff(fact_4035_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_4036_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_complex(semiri8010041392384452111omplex,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n1201886186963655149omplex((P)) ).

% of_nat_of_bool
tff(fact_4037_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_rat(semiri681578069525770553at_rat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n2052037380579107095ol_rat((P)) ).

% of_nat_of_bool
tff(fact_4038_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_4039_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_4040_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_Code_integer(zero_n356916108424825756nteger,(P)) ).

% of_nat_of_bool
tff(fact_4041_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mi3: nat,Ma: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Dega,TreeLista,Summarya)),X)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( X = Mi3 )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_4042_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_4043_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_4044_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_4045_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_4046_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4047_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_4048_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_4049_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_4050_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_4051_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4052_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_4053_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_4054_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_4055_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_4056_diff__numeral__special_I12_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4057_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4058_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4059_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4060_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4061_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(Na)) = uminus_uminus_rat(one_one_rat) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4062_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4063_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4064_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4065_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4066_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4067_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4068_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4069_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4070_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4071_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)) = one_one_real ).

% minus_one_mult_self
tff(fact_4072_minus__one__mult__self,axiom,
    ! [Na: 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)),Na)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)) = one_one_int ).

% minus_one_mult_self
tff(fact_4073_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)) = one_one_complex ).

% minus_one_mult_self
tff(fact_4074_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4075_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4076_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4077_left__minus__one__mult__self,axiom,
    ! [Na: 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)),Na)),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)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4078_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4079_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4080_left__minus__one__mult__self,axiom,
    ! [Na: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4081_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_4082_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4083_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_4084_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_4085_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_4086_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_4087_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_4088_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(plus_plus_num(V,W)))),Y) ).

% semiring_norm(168)
tff(fact_4089_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,plus_plus_num(V,W)))),Y) ).

% semiring_norm(168)
tff(fact_4090_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(plus_plus_num(V,W)))),Y) ).

% semiring_norm(168)
tff(fact_4091_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(plus_plus_num(V,W))),Y) ).

% semiring_norm(168)
tff(fact_4092_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(plus_plus_num(V,W)))),Y) ).

% semiring_norm(168)
tff(fact_4093_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_4094_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = numeral_numeral_real(plus_plus_num(M,Na)) ).

% diff_numeral_simps(2)
tff(fact_4095_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: 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,Na))) = aa_num_int(numeral_numeral_int,plus_plus_num(M,Na)) ).

% diff_numeral_simps(2)
tff(fact_4096_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(plus_plus_num(M,Na)) ).

% diff_numeral_simps(2)
tff(fact_4097_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(plus_plus_num(M,Na)) ).

% diff_numeral_simps(2)
tff(fact_4098_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(numeral_numeral_rat(Na))) = numeral_numeral_rat(plus_plus_num(M,Na)) ).

% diff_numeral_simps(2)
tff(fact_4099_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Na)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(plus_plus_num(M,Na))) ).

% diff_numeral_simps(3)
tff(fact_4100_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: 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,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,plus_plus_num(M,Na))) ).

% diff_numeral_simps(3)
tff(fact_4101_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(plus_plus_num(M,Na))) ).

% diff_numeral_simps(3)
tff(fact_4102_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(plus_plus_num(M,Na))) ).

% diff_numeral_simps(3)
tff(fact_4103_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(Na)) = uminus_uminus_rat(numeral_numeral_rat(plus_plus_num(M,Na))) ).

% diff_numeral_simps(3)
tff(fact_4104_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_4105_semiring__norm_I3_J,axiom,
    ! [Na: num] : plus_plus_num(one,aa_num_num(bit0,Na)) = aa_num_num(bit1,Na) ).

% semiring_norm(3)
tff(fact_4106_semiring__norm_I4_J,axiom,
    ! [Na: num] : plus_plus_num(one,aa_num_num(bit1,Na)) = aa_num_num(bit0,plus_plus_num(Na,one)) ).

% semiring_norm(4)
tff(fact_4107_semiring__norm_I5_J,axiom,
    ! [M: num] : plus_plus_num(aa_num_num(bit0,M),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_4108_semiring__norm_I8_J,axiom,
    ! [M: num] : plus_plus_num(aa_num_num(bit1,M),one) = aa_num_num(bit0,plus_plus_num(M,one)) ).

% semiring_norm(8)
tff(fact_4109_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = aa_num_num(bit0,plus_plus_num(plus_plus_num(M,Na),one)) ).

% semiring_norm(10)
tff(fact_4110_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(times_times_num(V,W))),Y) ).

% semiring_norm(172)
tff(fact_4111_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,times_times_num(V,W))),Y) ).

% semiring_norm(172)
tff(fact_4112_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(times_times_num(V,W))),Y) ).

% semiring_norm(172)
tff(fact_4113_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(times_times_num(V,W))),Y) ).

% semiring_norm(172)
tff(fact_4114_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(times_times_num(V,W))),Y) ).

% semiring_norm(172)
tff(fact_4115_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(times_times_num(V,W)))),Y) ).

% semiring_norm(171)
tff(fact_4116_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,times_times_num(V,W)))),Y) ).

% semiring_norm(171)
tff(fact_4117_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(times_times_num(V,W)))),Y) ).

% semiring_norm(171)
tff(fact_4118_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(times_times_num(V,W)))),Y) ).

% semiring_norm(171)
tff(fact_4119_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(times_times_num(V,W)))),Y) ).

% semiring_norm(171)
tff(fact_4120_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(times_times_num(V,W)))),Y) ).

% semiring_norm(170)
tff(fact_4121_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,times_times_num(V,W)))),Y) ).

% semiring_norm(170)
tff(fact_4122_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(times_times_num(V,W)))),Y) ).

% semiring_norm(170)
tff(fact_4123_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(times_times_num(V,W)))),Y) ).

% semiring_norm(170)
tff(fact_4124_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(times_times_num(V,W)))),Y) ).

% semiring_norm(170)
tff(fact_4125_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4126_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: 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,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,times_times_num(M,Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4127_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4128_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4129_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Na))) = uminus_uminus_rat(numeral_numeral_rat(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4130_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Na)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4131_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: 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,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,times_times_num(M,Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4132_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4133_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4134_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(Na)) = uminus_uminus_rat(numeral_numeral_rat(times_times_num(M,Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4135_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = numeral_numeral_real(times_times_num(M,Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4136_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: 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,Na))) = aa_num_int(numeral_numeral_int,times_times_num(M,Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4137_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(times_times_num(M,Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4138_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(times_times_num(M,Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4139_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Na))) = numeral_numeral_rat(times_times_num(M,Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4140_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)))
    <=> aa_num_o(ord_less_eq_num(Na),M) ) ).

% neg_numeral_le_iff
tff(fact_4141_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Na)))
    <=> aa_num_o(ord_less_eq_num(Na),M) ) ).

% neg_numeral_le_iff
tff(fact_4142_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(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,Na)))
    <=> aa_num_o(ord_less_eq_num(Na),M) ) ).

% neg_numeral_le_iff
tff(fact_4143_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)))
    <=> aa_num_o(ord_less_eq_num(Na),M) ) ).

% neg_numeral_le_iff
tff(fact_4144_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)))
    <=> aa_num_o(ord_less_num(Na),M) ) ).

% neg_numeral_less_iff
tff(fact_4145_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(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,Na)))
    <=> aa_num_o(ord_less_num(Na),M) ) ).

% neg_numeral_less_iff
tff(fact_4146_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)))
    <=> aa_num_o(ord_less_num(Na),M) ) ).

% neg_numeral_less_iff
tff(fact_4147_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Na)))
    <=> aa_num_o(ord_less_num(Na),M) ) ).

% neg_numeral_less_iff
tff(fact_4148_semiring__norm_I16_J,axiom,
    ! [M: num,Na: num] : times_times_num(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = aa_num_num(bit1,plus_plus_num(plus_plus_num(M,Na),aa_num_num(bit0,times_times_num(M,Na)))) ).

% semiring_norm(16)
tff(fact_4149_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(74)
tff(fact_4150_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(79)
tff(fact_4151_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4152_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4153_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(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_4154_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4155_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4156_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4157_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4158_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4159_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4160_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))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4161_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4162_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4163_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 )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4164_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) = A )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4165_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4166_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(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_4167_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4168_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4169_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4170_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4171_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4172_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4173_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_4174_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_4175_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_4176_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_4177_power2__minus,axiom,
    ! [A: real] : aa_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4178_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_4179_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_4180_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_4181_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_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_4182_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4183_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4184_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4185_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4186_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4187_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4188_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4189_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4190_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4191_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4192_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4193_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4194_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4195_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4196_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4197_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(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_4198_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(power_power_real(numeral_numeral_real(X)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4199_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4200_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,Na: 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)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4201_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4202_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_4203_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: nat] :
      ( ( aa_nat_real(power_power_real(numeral_numeral_real(X)),Na) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_4204_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_4205_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_4206_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),Na) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_4207_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_4208_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),numeral_numeral_real(W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_4209_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_4210_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Na)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_4211_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4212_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_4213_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_4214_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_4215_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4216_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4217_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_4218_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_4219_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_4220_diff__numeral__special_I10_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4221_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4222_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_4223_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_4224_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_4225_diff__numeral__special_I11_J,axiom,
    minus_minus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4226_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_4227_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_4228_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_4229_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_4230_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_4231_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_4232_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4233_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4234_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4235_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_4236_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Na: nat] : aa_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))),Na)) = aa_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))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4237_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Na: 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))),Na)) = 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))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4238_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Na: 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))),Na)) = 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))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4239_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Na: 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))),Na)) = 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))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4240_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = 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))),Na)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4241_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na) = aa_nat_real(power_power_real(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4242_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = aa_nat_int(power_power_int(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4243_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Na) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4244_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4245_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4246_power__minus__odd,axiom,
    ! [Na: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na) = aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(A),Na)) ) ) ).

% power_minus_odd
tff(fact_4247_power__minus__odd,axiom,
    ! [Na: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_minus_odd
tff(fact_4248_power__minus__odd,axiom,
    ! [Na: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Na) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Na)) ) ) ).

% power_minus_odd
tff(fact_4249_power__minus__odd,axiom,
    ! [Na: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ) ) ).

% power_minus_odd
tff(fact_4250_power__minus__odd,axiom,
    ! [Na: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_minus_odd
tff(fact_4251_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(plus_plus_num(M,one))) ).

% diff_numeral_special(4)
tff(fact_4252_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,plus_plus_num(M,one))) ).

% diff_numeral_special(4)
tff(fact_4253_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(plus_plus_num(M,one))) ).

% diff_numeral_special(4)
tff(fact_4254_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(plus_plus_num(M,one))) ).

% diff_numeral_special(4)
tff(fact_4255_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(plus_plus_num(M,one))) ).

% diff_numeral_special(4)
tff(fact_4256_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = numeral_numeral_real(plus_plus_num(one,Na)) ).

% diff_numeral_special(3)
tff(fact_4257_diff__numeral__special_I3_J,axiom,
    ! [Na: 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,Na))) = aa_num_int(numeral_numeral_int,plus_plus_num(one,Na)) ).

% diff_numeral_special(3)
tff(fact_4258_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(plus_plus_num(one,Na)) ).

% diff_numeral_special(3)
tff(fact_4259_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(plus_plus_num(one,Na)) ).

% diff_numeral_special(3)
tff(fact_4260_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(Na))) = numeral_numeral_rat(plus_plus_num(one,Na)) ).

% diff_numeral_special(3)
tff(fact_4261_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_4262_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: 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,Na)))) = 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))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_4263_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_4264_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: 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,Na)))) = 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))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_4265_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),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(Na),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_4266_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4267_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_4268_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_4269_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_4270_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4271_power__minus1__even,axiom,
    ! [Na: nat] : aa_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))),Na)) = one_one_real ).

% power_minus1_even
tff(fact_4272_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_int ).

% power_minus1_even
tff(fact_4273_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_complex ).

% power_minus1_even
tff(fact_4274_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_4275_power__minus1__even,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = one_one_rat ).

% power_minus1_even
tff(fact_4276_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_4277_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_4278_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_4279_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_4280_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_4281_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_4282_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_4283_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_4284_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_4285_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_4286_even__of__nat,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_of_nat
tff(fact_4287_even__of__nat,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_of_nat
tff(fact_4288_even__of__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_of_nat
tff(fact_4289_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(numeral_numeral_real(I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4290_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4291_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4292_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4293_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(numeral_numeral_real(I)),Na)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4294_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4295_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4296_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Na)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4297_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4298_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4299_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4300_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(numeral_numeral_real(I)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4301_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Na)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4302_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4303_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Na)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4304_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(numeral_numeral_real(I)),Na)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Na)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4305_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_4306_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_4307_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(power_power_real(aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% real_sqrt_pow2_iff
tff(fact_4308_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_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_4309_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : aa_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_4310_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_4311_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),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(Na),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_4312_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),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(Na),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_4313_take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(Na),K))) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_4314_signed__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(Na),K))) = aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_4315_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_4316_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_4317_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_4318_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_4319_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( uminus_uminus_rat(B) = A ) ) ).

% minus_equation_iff
tff(fact_4320_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_4321_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_4322_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_4323_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_4324_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% equation_minus_iff
tff(fact_4325_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_4326_power__minus__Bit1,axiom,
    ! [X: real,K: num] : aa_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4327_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_4328_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_4329_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_4330_power__minus__Bit1,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4331_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_4332_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_4333_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_4334_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_4335_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_4336_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_4337_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_4338_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_4339_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_4340_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa_real_real(sqrt,aa_nat_real(power_power_real(X),K)) = aa_nat_real(power_power_real(aa_real_real(sqrt,X)),K) ).

% real_sqrt_power
tff(fact_4341_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) )
       => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_4342_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) )
       => ( ! [A4: product_prod_nat_nat,B4: num] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_num(B4) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_4343_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) )
       => ( ! [A4: num,B4: product_prod_nat_nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_4344_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( X = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) )
       => ( ! [A4: num,B4: num] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_num(B4) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_4345_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_all
tff(fact_4346_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_all
tff(fact_4347_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_ex
tff(fact_4348_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_4349_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_4350_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_4351_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_4352_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_4353_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_4354_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_4355_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_4356_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% le_imp_neg_le
tff(fact_4357_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_4358_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_4359_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_4360_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),A) ) ).

% minus_le_iff
tff(fact_4361_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_4362_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_4363_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_4364_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),uminus_uminus_rat(A)) ) ).

% le_minus_iff
tff(fact_4365_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_4366_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_4367_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4368_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4369_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4370_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4371_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_4372_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_4373_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_4374_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),A) ) ).

% minus_less_iff
tff(fact_4375_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_4376_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_4377_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_4378_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),uminus_uminus_rat(A)) ) ).

% less_minus_iff
tff(fact_4379_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_4380_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_4381_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : numeral_numeral_real(M) != aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)) ).

% numeral_neq_neg_numeral
tff(fact_4382_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% numeral_neq_neg_numeral
tff(fact_4383_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% numeral_neq_neg_numeral
tff(fact_4384_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : numera6620942414471956472nteger(M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% numeral_neq_neg_numeral
tff(fact_4385_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : numeral_numeral_rat(M) != uminus_uminus_rat(numeral_numeral_rat(Na)) ).

% numeral_neq_neg_numeral
tff(fact_4386_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) != numeral_numeral_real(Na) ).

% neg_numeral_neq_numeral
tff(fact_4387_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,Na) ).

% neg_numeral_neq_numeral
tff(fact_4388_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(Na) ).

% neg_numeral_neq_numeral
tff(fact_4389_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(Na) ).

% neg_numeral_neq_numeral
tff(fact_4390_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : uminus_uminus_rat(numeral_numeral_rat(M)) != numeral_numeral_rat(Na) ).

% neg_numeral_neq_numeral
tff(fact_4391_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_4392_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_4393_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_4394_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_4395_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) ).

% minus_mult_commute
tff(fact_4396_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_4397_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_4398_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_4399_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_4400_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = uminus_uminus_rat(B) ) ) ) ).

% square_eq_iff
tff(fact_4401_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_4402_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_4403_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_4404_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_4405_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_4406_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_4407_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_4408_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_4409_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_4410_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ).

% add.inverse_distrib_swap
tff(fact_4411_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4412_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = 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_4413_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_4414_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4415_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( uminus_uminus_rat(A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(K)),uminus_uminus_rat(A)) ) ) ).

% group_cancel.neg1
tff(fact_4416_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_4417_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_4418_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_4419_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_4420_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ).

% is_num_normalize(8)
tff(fact_4421_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_4422_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_4423_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_4424_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_4425_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : minus_minus_rat(uminus_uminus_rat(B),A) = minus_minus_rat(uminus_uminus_rat(A),B) ).

% minus_diff_commute
tff(fact_4426_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_4427_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_4428_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_4429_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_4430_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(minus_minus_rat(A,B)) ).

% minus_diff_minus
tff(fact_4431_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_4432_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_4433_minus__divide__right,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ).

% minus_divide_right
tff(fact_4434_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_4435_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_4436_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ).

% minus_divide_divide
tff(fact_4437_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_4438_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_4439_minus__divide__left,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(uminus_uminus_rat(A),B) ).

% minus_divide_left
tff(fact_4440_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_4441_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_4442_take__bit__of__nat,axiom,
    ! [Na: nat,M: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)) ).

% take_bit_of_nat
tff(fact_4443_take__bit__of__nat,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)) ).

% take_bit_of_nat
tff(fact_4444_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_4445_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_4446_mod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A3),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% mod_minus_cong
tff(fact_4447_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3),B) ) ) ).

% mod_minus_cong
tff(fact_4448_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_4449_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_4450_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_4451_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),zero_zero_nat,Va2,Vb)),X)
    <=> ( ( X = Mi3 )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_4452_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_gt_zero
tff(fact_4453_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_4454_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_ge_zero
tff(fact_4455_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_0_le_iff
tff(fact_4456_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_0_le_iff
tff(fact_4457_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_0_le_iff
tff(fact_4458_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_0_le_iff
tff(fact_4459_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_ge_one
tff(fact_4460_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_4461_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_4462_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_4463_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_4464_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_4465_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_4466_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_4467_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Na)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_4468_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_4469_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_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,Na))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% div_mult2_eq'
tff(fact_4470_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,Na: 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,Na))) = 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,Na)) ).

% div_mult2_eq'
tff(fact_4471_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_4472_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_4473_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_4474_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_4475_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% less_imp_of_nat_less
tff(fact_4476_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% less_imp_of_nat_less
tff(fact_4477_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_4478_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_4479_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_4480_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_4481_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_4482_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_4483_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_numeral_le_neg_numeral
tff(fact_4484_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Na))) ).

% not_numeral_le_neg_numeral
tff(fact_4485_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_4486_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) ).

% not_numeral_le_neg_numeral
tff(fact_4487_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Na)) ).

% neg_numeral_le_numeral
tff(fact_4488_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(Na)) ).

% neg_numeral_le_numeral
tff(fact_4489_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_le_numeral
tff(fact_4490_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Na)) ).

% neg_numeral_le_numeral
tff(fact_4491_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)) ).

% zero_neq_neg_numeral
tff(fact_4492_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% zero_neq_neg_numeral
tff(fact_4493_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% zero_neq_neg_numeral
tff(fact_4494_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% zero_neq_neg_numeral
tff(fact_4495_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(Na)) ).

% zero_neq_neg_numeral
tff(fact_4496_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Na)) ).

% neg_numeral_less_numeral
tff(fact_4497_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_less_numeral
tff(fact_4498_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Na)) ).

% neg_numeral_less_numeral
tff(fact_4499_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(Na)) ).

% neg_numeral_less_numeral
tff(fact_4500_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) ).

% not_numeral_less_neg_numeral
tff(fact_4501_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_4502_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_numeral_less_neg_numeral
tff(fact_4503_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Na))) ).

% not_numeral_less_neg_numeral
tff(fact_4504_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_4505_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_4506_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_4507_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_4508_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_4509_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_4510_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_4511_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_4512_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_4513_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_4514_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_4515_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_4516_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_4517_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_4518_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_4519_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_4520_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_4521_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4522_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_4523_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_4524_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_4525_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_4526_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4527_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_4528_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_4529_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_4530_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_4531_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( uminus_uminus_rat(A) = B ) ) ).

% add.inverse_unique
tff(fact_4532_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_4533_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_4534_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_4535_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_4536_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_4537_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_4538_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_4539_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_4540_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_4541_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% add_eq_0_iff
tff(fact_4542_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_4543_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_4544_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_4545_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_4546_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_4547_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_4548_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_4549_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_4550_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_4551_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% numeral_times_minus_swap
tff(fact_4552_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_4553_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_4554_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_4555_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),uminus_uminus_rat(X)) = aa_rat_rat(times_times_rat(X),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_4556_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_4557_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_4558_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4559_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_4560_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_4561_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4562_numeral__neq__neg__one,axiom,
    ! [Na: num] : numeral_numeral_real(Na) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_4563_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,Na) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_4564_numeral__neq__neg__one,axiom,
    ! [Na: num] : numera6690914467698888265omplex(Na) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_4565_numeral__neq__neg__one,axiom,
    ! [Na: num] : numera6620942414471956472nteger(Na) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_4566_numeral__neq__neg__one,axiom,
    ! [Na: num] : numeral_numeral_rat(Na) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_4567_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Na)) ).

% one_neq_neg_numeral
tff(fact_4568_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% one_neq_neg_numeral
tff(fact_4569_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% one_neq_neg_numeral
tff(fact_4570_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na)) ).

% one_neq_neg_numeral
tff(fact_4571_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(Na)) ).

% one_neq_neg_numeral
tff(fact_4572_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_4573_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_4574_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_4575_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_4576_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_4577_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_4578_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_4579_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_4580_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_4581_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_4582_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_4583_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4584_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_4585_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_4586_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_4587_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_4588_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(B)) ).

% diff_conv_add_uminus
tff(fact_4589_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4590_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = 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),B2) = 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_4591_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4592_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_4593_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( minus_minus_rat(A,B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(K)),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_4594_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_4595_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_4596_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_4597_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_4598_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_4599_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_4600_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_4601_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(A,uminus_uminus_rat(B)) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_div_neg
tff(fact_4602_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_4603_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_4604_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_4605_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_4606_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(uminus_uminus_rat(A),B) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_neg_div
tff(fact_4607_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_mod
tff(fact_4608_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mod
tff(fact_4609_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mod
tff(fact_4610_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : aa_real_o(aa_real_fun_real_o(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_4611_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = 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_4612_zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( Na = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Na = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_4613_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_4614_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_4615_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_4616_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_4617_Bernoulli__inequality,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(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,Na)),X))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),Na)) ) ).

% Bernoulli_inequality
tff(fact_4618_real__div__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_4619_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(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_4620_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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_4621_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_zero_le_neg_numeral
tff(fact_4622_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(Na))) ).

% not_zero_le_neg_numeral
tff(fact_4623_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_zero_le_neg_numeral
tff(fact_4624_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) ).

% not_zero_le_neg_numeral
tff(fact_4625_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_4626_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Na))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_4627_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_4628_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_4629_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) ).

% not_zero_less_neg_numeral
tff(fact_4630_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_zero_less_neg_numeral
tff(fact_4631_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) ).

% not_zero_less_neg_numeral
tff(fact_4632_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(Na))) ).

% not_zero_less_neg_numeral
tff(fact_4633_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_4634_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_4635_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_4636_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Na))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_4637_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_4638_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_4639_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_4640_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_4641_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_4642_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_4643_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_4644_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_4645_numeral__Bit1,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit1,Na)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),numera6690914467698888265omplex(Na))),one_one_complex) ).

% numeral_Bit1
tff(fact_4646_numeral__Bit1,axiom,
    ! [Na: num] : numeral_numeral_real(aa_num_num(bit1,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Na)),numeral_numeral_real(Na))),one_one_real) ).

% numeral_Bit1
tff(fact_4647_numeral__Bit1,axiom,
    ! [Na: num] : numeral_numeral_rat(aa_num_num(bit1,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Na)),numeral_numeral_rat(Na))),one_one_rat) ).

% numeral_Bit1
tff(fact_4648_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = 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,Na)),aa_num_nat(numeral_numeral_nat,Na))),one_one_nat) ).

% numeral_Bit1
tff(fact_4649_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = 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,Na)),aa_num_int(numeral_numeral_int,Na))),one_one_int) ).

% numeral_Bit1
tff(fact_4650_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_4651_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_4652_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_4653_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_4654_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_4655_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_4656_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_4657_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_4658_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% of_nat_diff
tff(fact_4659_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ) ).

% of_nat_diff
tff(fact_4660_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ) ).

% of_nat_diff
tff(fact_4661_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Na)) ) ) ).

% of_nat_diff
tff(fact_4662_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ) ).

% of_nat_diff
tff(fact_4663_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))) ).

% not_one_le_neg_numeral
tff(fact_4664_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_one_le_neg_numeral
tff(fact_4665_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(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_4666_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_one_le_neg_numeral
tff(fact_4667_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_4668_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_4669_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(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_4670_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_4671_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_4672_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_4673_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(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_4674_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_4675_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M)) ).

% neg_one_le_numeral
tff(fact_4676_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_le_numeral
tff(fact_4677_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(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_4678_neg__one__le__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M)) ).

% neg_one_le_numeral
tff(fact_4679_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_4680_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_4681_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(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_4682_neg__numeral__le__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) ).

% neg_numeral_le_one
tff(fact_4683_neg__numeral__less__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) ).

% neg_numeral_less_one
tff(fact_4684_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(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_4685_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_4686_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_4687_neg__one__less__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M)) ).

% neg_one_less_numeral
tff(fact_4688_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(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_4689_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M)) ).

% neg_one_less_numeral
tff(fact_4690_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_less_numeral
tff(fact_4691_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_4692_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(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_4693_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_4694_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_4695_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_one_less_neg_numeral
tff(fact_4696_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(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_4697_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))) ).

% not_one_less_neg_numeral
tff(fact_4698_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_one_less_neg_numeral
tff(fact_4699_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4700_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(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_4701_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4702_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4703_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_4704_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_4705_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = uminus_uminus_rat(divide_divide_rat(A,B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = uminus_uminus_rat(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4706_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_4707_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_4708_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat(divide_divide_rat(A,B)) = C )
      <=> ( uminus_uminus_rat(A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4709_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 )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_4710_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_4711_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( uminus_uminus_rat(divide_divide_rat(B,C)) = A )
    <=> $ite(C != zero_zero_rat,uminus_uminus_rat(B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_4712_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)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_4713_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_4714_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = uminus_uminus_rat(divide_divide_rat(B,C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = uminus_uminus_rat(B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_4715_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_4716_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_4717_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = uminus_uminus_rat(one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = uminus_uminus_rat(B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4718_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_4719_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_4720_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_4721_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_4722_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(1)
tff(fact_4723_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_4724_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_4725_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_4726_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_4727_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(2)
tff(fact_4728_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_4729_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_4730_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_4731_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_4732_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_4733_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ).

% eval_nat_numeral(3)
tff(fact_4734_power__minus,axiom,
    ! [A: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(power_power_real(A),Na)) ).

% power_minus
tff(fact_4735_power__minus,axiom,
    ! [A: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = 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)),Na)),aa_nat_int(power_power_int(A),Na)) ).

% power_minus
tff(fact_4736_power__minus,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(power_power_complex(A),Na)) ).

% power_minus
tff(fact_4737_power__minus,axiom,
    ! [A: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% power_minus
tff(fact_4738_power__minus,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)),aa_nat_rat(power_power_rat(A),Na)) ).

% power_minus
tff(fact_4739_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_4740_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_4741_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_4742_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_4743_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_4744_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,Na: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_4745_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),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,Na)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4746_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,Na: 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,Na))),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,Na)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4747_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,Na: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4748_power__minus__Bit0,axiom,
    ! [X: real,K: num] : aa_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_4749_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_4750_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_4751_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_4752_power__minus__Bit0,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_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_4753_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ! [Y3: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% reals_Archimedean3
tff(fact_4754_real__of__nat__div4,axiom,
    ! [Na: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,X))) ).

% real_of_nat_div4
tff(fact_4755_real__of__nat__div,axiom,
    ! [D: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_4756_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_less_0_iff
tff(fact_4757_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y) ) ).

% real_0_less_add_iff
tff(fact_4758_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_le_0_iff
tff(fact_4759_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y) ) ).

% real_0_le_add_iff
tff(fact_4760_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,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_4761_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,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_4762_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4763_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(some_num(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4764_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_4765_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Na: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) = 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,Na))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_4766_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,Na: 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,Na))) = 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(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Na)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_4767_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,Na: 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,Na))) = 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,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_4768_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Na) ).

% numeral_Bit1_div_2
tff(fact_4769_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Na) ).

% numeral_Bit1_div_2
tff(fact_4770_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(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_4771_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4772_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(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_4773_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4774_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(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_4775_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4776_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(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_4777_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4778_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_4779_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_4780_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_4781_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_4782_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) = B,aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4783_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) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4784_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) = B,uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4785_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4786_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)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4787_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C),uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4788_odd__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,Na))) ).

% odd_numeral
tff(fact_4789_odd__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))) ).

% odd_numeral
tff(fact_4790_odd__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))) ).

% odd_numeral
tff(fact_4791_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: 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) = $ite(Z = zero_zero_real,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_4792_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,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_4793_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(divide_divide_rat(A,Z))),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4794_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_4795_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_4796_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(divide_divide_rat(X,Z))),Y) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(X)),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4797_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_4798_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_4799_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_4800_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: 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) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,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_4801_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,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_4802_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4803_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,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_4804_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,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_4805_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(divide_divide_rat(A,Z),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),divide_divide_rat(minus_minus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4806_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_4807_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_4808_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4809_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_4810_power3__eq__cube,axiom,
    ! [A: real] : aa_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_4811_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_4812_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_4813_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_4814_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_4815_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_4816_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_4817_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_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_4818_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_4819_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_4820_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_4821_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 = uminus_uminus_rat(Y) ) ) ) ).

% power2_eq_iff
tff(fact_4822_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na) ).

% Suc3_eq_add_3
tff(fact_4823_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = 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),Na)))),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_4824_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = 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),Na)))),aa_nat_complex(semiri8010041392384452111omplex,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_4825_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_4826_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_4827_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_4828_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_4829_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_4830_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_4831_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_4832_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_4833_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_4834_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_4835_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_4836_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_4837_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_4838_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_4839_inverse__of__nat__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( Na != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_4840_inverse__of__nat__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( Na != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(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,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_4841_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_4842_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_4843_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4844_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4845_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4846_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(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_4847_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4848_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(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_4849_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4850_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(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_4851_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4852_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(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_4853_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4854_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4855_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4856_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4857_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Na: 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,Na))),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,Na)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4858_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Na: 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,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4859_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Na: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Na)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Na),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4860_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_4861_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_4862_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_4863_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_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_4864_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_4865_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_4866_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_4867_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 = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_4868_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M4)
             => aa_real_o(aa_real_fun_real_o(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_4869_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: 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)))),Na) = 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)),Na) ).

% Suc_div_eq_add3_div
tff(fact_4870_uminus__power__if,axiom,
    ! [A: real,Na: nat] :
      aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_real(power_power_real(A),Na),aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(A),Na))) ).

% uminus_power_if
tff(fact_4871_uminus__power__if,axiom,
    ! [A: int,Na: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_int(power_power_int(A),Na),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Na))) ).

% uminus_power_if
tff(fact_4872_uminus__power__if,axiom,
    ! [A: complex,Na: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_complex(power_power_complex(A),Na),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Na))) ).

% uminus_power_if
tff(fact_4873_uminus__power__if,axiom,
    ! [A: code_integer,Na: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_Code_integer(power_8256067586552552935nteger(A),Na),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na))) ).

% uminus_power_if
tff(fact_4874_uminus__power__if,axiom,
    ! [A: rat,Na: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_rat(power_power_rat(A),Na),uminus_uminus_rat(aa_nat_rat(power_power_rat(A),Na))) ).

% uminus_power_if
tff(fact_4875_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_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,Na),K)) = aa_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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4876_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4877_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4878_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4879_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4880_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: 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)))),Na) = 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)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_4881_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_4882_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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(Na),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_4883_real__of__nat__div2,axiom,
    ! [Na: nat,X: nat] : aa_real_o(aa_real_fun_real_o(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,Na)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),X)))) ).

% real_of_nat_div2
tff(fact_4884_real__of__nat__div3,axiom,
    ! [Na: nat,X: nat] : aa_real_o(aa_real_fun_real_o(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,Na)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),X)))),one_one_real) ).

% real_of_nat_div3
tff(fact_4885_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(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))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_4886_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(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))),Na))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_4887_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(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))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_4888_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(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_4889_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(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_4890_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_4891_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_4892_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_4893_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_4894_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_4895_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_4896_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_4897_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_4898_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_4899_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_4900_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4901_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4902_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4903_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4904_square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_4905_square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(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_4906_square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),one_one_int)
       => aa_int_o(aa_int_fun_int_o(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_4907_square__le__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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_4908_minus__power__mult__self,axiom,
    ! [A: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na)) = aa_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))),Na)) ).

% minus_power_mult_self
tff(fact_4909_minus__power__mult__self,axiom,
    ! [A: int,Na: 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)),Na)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na)) = 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))),Na)) ).

% minus_power_mult_self
tff(fact_4910_minus__power__mult__self,axiom,
    ! [A: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Na)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Na)) = 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))),Na)) ).

% minus_power_mult_self
tff(fact_4911_minus__power__mult__self,axiom,
    ! [A: code_integer,Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na)) = 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))),Na)) ).

% minus_power_mult_self
tff(fact_4912_minus__power__mult__self,axiom,
    ! [A: rat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na)) = 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))),Na)) ).

% minus_power_mult_self
tff(fact_4913_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_4914_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_4915_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_4916_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_4917_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_rat,uminus_uminus_rat(one_one_rat)) ).

% minus_one_power_iff
tff(fact_4918_linear__plus__1__le__power,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(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,Na)),X)),one_one_real)),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),Na)) ) ).

% linear_plus_1_le_power
tff(fact_4919_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(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))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_4920_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(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))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_4921_minus__1__div__exp__eq__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_4922_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_4923_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_4924_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_4925_sqrt__even__pow2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_real(sqrt,aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),Na)) = aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_4926_power__minus1__odd,axiom,
    ! [Na: nat] : aa_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))),Na))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_4927_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_4928_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_4929_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_4930_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_4931_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K3: int] :
              ( aa_int_o(P,K3)
             => ( ( K3 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K3: int] :
                ( aa_int_o(P,K3)
               => ( ( K3 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(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,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_4932_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_4933_real__sqrt__power__even,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X)),Na) = aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_4934_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_4935_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,M: nat,Dega: nat,Mi3: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(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] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mi3 = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(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))),Dega))
                     => ( ( ( Mi3 != Ma )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(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 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Na)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Na) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_4936_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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(Na),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_4937_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),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,Na),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_4938_take__bit__Suc__minus__1__eq,axiom,
    ! [Na: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Na),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,Na)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_4939_take__bit__Suc__minus__1__eq,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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,Na))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_4940_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_4941_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_4942_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Na: nat,Summarya: vEBT_VEBT,M: nat,Dega: nat,Mi3: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = 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,Na) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(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] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mi3 = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(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))),Dega))
                     => ( ( ( Mi3 != Ma )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(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 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Na)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Na) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_4943_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))),Na)))
     => aa_int_o(aa_int_fun_int_o(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,Na)))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_4944_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_se2923211474154528505it_int(Na),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))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4945_member__inv,axiom,
    ! [Mi3: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Dega,TreeLista,Summarya)),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( X = Mi3 )
          | ( X = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X)
            & aa_nat_o(aa_nat_fun_nat_o(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_4946_divmod__step__eq,axiom,
    ! [L: num,Q3: nat,R4: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R4)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R4),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,R4),aa_num_nat(numeral_numeral_nat,L))),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)),R4)) ).

% divmod_step_eq
tff(fact_4947_divmod__step__eq,axiom,
    ! [L: num,Q3: int,R4: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q3,R4)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R4),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,R4),aa_num_int(numeral_numeral_int,L))),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),R4)) ).

% divmod_step_eq
tff(fact_4948_divmod__step__eq,axiom,
    ! [L: num,Q3: code_integer,R4: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R4)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R4),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R4,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),R4)) ).

% divmod_step_eq
tff(fact_4949_arsinh__real__aux,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_4950_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_4951_nat__approx__posE,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_4952_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_4953_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Y),X) ) ).

% compl_le_compl_iff
tff(fact_4954_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_4955_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_4956_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_4957_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_4958_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_4959_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_4960_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_4961_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_4962_int__dvd__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% int_dvd_int_iff
tff(fact_4963_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_4964_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_4965_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_4966_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_4967_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_4968_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_4969_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_4970_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_4971_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_4972_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_4973_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% int_cases2
tff(fact_4974_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_4975_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_4976_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_4977_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M4)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_4978_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_4979_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_4980_int__ops_I3_J,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% int_ops(3)
tff(fact_4981_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_4982_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_4983_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% zle_int
tff(fact_4984_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_4985_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_4986_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_4987_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_4988_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_4989_zadd__int__left,axiom,
    ! [M: nat,Na: 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,Na)),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),Na))),Z) ).

% zadd_int_left
tff(fact_4990_int__plus,axiom,
    ! [Na: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_4991_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_4992_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_4993_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_4994_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_4995_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_4996_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D3)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_4997_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_4998_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M4: num] : X != product_Pair_num_num(aa_num_num(bit0,M4),one)
           => ( ! [M4: num,N: num] : X != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N))
             => ( ! [M4: num,N: num] : X != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N))
               => ( ! [M4: num] : X != product_Pair_num_num(aa_num_num(bit1,M4),one)
                 => ( ! [M4: num,N: num] : X != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N))
                   => ~ ! [M4: num,N: num] : X != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_4999_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))) ).

% not_zle_0_negative
tff(fact_5000_negD,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
     => ? [N: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_5001_negative__zless__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_5002_int__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ).

% int_Suc
tff(fact_5003_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_5004_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_5005_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X) ).

% vebt_member.simps(2)
tff(fact_5006_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_5007_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_5008_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_5009_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(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_5010_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,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_5011_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X2)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_5012_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),X)
    <=> $ite(
          X = zero_zero_nat,
          (A),
          $ite(X = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_5013_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X) ).

% vebt_member.simps(3)
tff(fact_5014_zdiff__int__split,axiom,
    ! [P: fun_int_o,X: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_5015_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X) ).

% vebt_member.simps(4)
tff(fact_5016_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_5017_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_5018_dbl__dec__def,axiom,
    ! [X: rat] : neg_nu3179335615603231917ec_rat(X) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X),one_one_rat) ).

% dbl_dec_def
tff(fact_5019_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_5020_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X)) ) ).

% compl_mono
tff(fact_5021_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X))
     => aa_set_nat_o(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_5022_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y) ) ).

% compl_le_swap2
tff(fact_5023_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_5024_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_5025_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_5026_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_5027_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_5028_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z) ) ) ).

% eq_diff_eq'
tff(fact_5029_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2),X2)
         => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),X2)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_5030_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X2)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X2)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X2)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X2) ) ) ) ) ).

% vebt_member.cases
tff(fact_5031_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_5032_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_5033_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_5034_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(power_power_real(Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5035_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Na: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(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)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5036_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_5037_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(ord_less_num(M),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,Na),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_5038_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(ord_less_num(M),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),unique5024387138958732305ep_int(aa_num_num(bit1,Na),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_5039_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(ord_less_num(M),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_5040_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(ord_less_eq_num(M),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,Na),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_5041_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(ord_less_eq_num(M),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),unique5024387138958732305ep_int(aa_num_num(bit1,Na),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_5042_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(ord_less_eq_num(M),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_5043_divides__aux__eq,axiom,
    ! [Q3: nat,R4: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R4))
    <=> ( R4 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_5044_divides__aux__eq,axiom,
    ! [Q3: int,R4: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q3,R4))
    <=> ( R4 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_5045_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5046_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5047_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa_nat_nat(suc,Na) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Na = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_5048_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Na) )
    <=> ( aa_num_nat(pred_numeral,K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_5049_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_5050_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% less_numeral_Suc
tff(fact_5051_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_5052_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% le_numeral_Suc
tff(fact_5053_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_5054_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Na) ).

% diff_numeral_Suc
tff(fact_5055_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_5056_dvd__numeral__simp,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> unique6319869463603278526ux_int(unique5052692396658037445od_int(Na,M)) ) ).

% dvd_numeral_simp
tff(fact_5057_dvd__numeral__simp,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> unique6322359934112328802ux_nat(unique5055182867167087721od_nat(Na,M)) ) ).

% dvd_numeral_simp
tff(fact_5058_dvd__numeral__simp,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Na))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Na,M)) ) ).

% dvd_numeral_simp
tff(fact_5059_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_5060_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_5061_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5062_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Na)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5063_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5064_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5065_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Na)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5066_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5067_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5068_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_5069_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_5070_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_5071_int__int__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% int_int_eq
tff(fact_5072_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_5073_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_5074_divmod__int__def,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(M,Na) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))) ).

% divmod_int_def
tff(fact_5075_divmod__def,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(M,Na) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))) ).

% divmod_def
tff(fact_5076_divmod__def,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(M,Na) = 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))) ).

% divmod_def
tff(fact_5077_divmod__def,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(M,Na) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Na)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Na))) ).

% divmod_def
tff(fact_5078_divmod_H__nat__def,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(M,Na) = 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))) ).

% divmod'_nat_def
tff(fact_5079_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_5080_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_5081_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_5082_divmod__divmod__step,axiom,
    ! [M: num,Na: num] :
      unique5055182867167087721od_nat(M,Na) = $ite(aa_num_o(ord_less_num(M),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),unique5026877609467782581ep_nat(Na,unique5055182867167087721od_nat(M,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_5083_divmod__divmod__step,axiom,
    ! [M: num,Na: num] :
      unique5052692396658037445od_int(M,Na) = $ite(aa_num_o(ord_less_num(M),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)),unique5024387138958732305ep_int(Na,unique5052692396658037445od_int(M,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_5084_divmod__divmod__step,axiom,
    ! [M: num,Na: num] :
      unique3479559517661332726nteger(M,Na) = $ite(aa_num_o(ord_less_num(M),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)),unique4921790084139445826nteger(Na,unique3479559517661332726nteger(M,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_5085_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_5086_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_5087_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5088_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5089_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_5090_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_5091_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5092_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5093_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_5094_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_5095_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real)
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5096_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5097_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_5098_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_5099_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5100_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5101_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_5102_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_5103_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_5104_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_5105_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_5106_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_5107_norm__not__less__zero,axiom,
    ! [X: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_5108_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_5109_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_5110_norm__power,axiom,
    ! [X: real,Na: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(power_power_real(X),Na)) = aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,X)),Na) ).

% norm_power
tff(fact_5111_norm__power,axiom,
    ! [X: complex,Na: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),Na)) = aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,X)),Na) ).

% norm_power
tff(fact_5112_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_5113_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_5114_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_5115_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_5116_power__eq__imp__eq__norm,axiom,
    ! [W: real,Na: nat,Z: real] :
      ( ( aa_nat_real(power_power_real(W),Na) = aa_nat_real(power_power_real(Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5117_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Na: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Na) = aa_nat_complex(power_power_complex(Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5118_norm__mult__less,axiom,
    ! [X: real,R4: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(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,R4),S)) ) ) ).

% norm_mult_less
tff(fact_5119_norm__mult__less,axiom,
    ! [X: complex,R4: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(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,R4),S)) ) ) ).

% norm_mult_less
tff(fact_5120_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => aa_real_o(aa_real_fun_real_o(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_5121_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_5122_norm__add__less,axiom,
    ! [X: real,R4: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(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,R4),S)) ) ) ).

% norm_add_less
tff(fact_5123_norm__add__less,axiom,
    ! [X: complex,R4: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(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,R4),S)) ) ) ).

% norm_add_less
tff(fact_5124_norm__triangle__mono,axiom,
    ! [A: real,R4: real,B: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S)
       => aa_real_o(aa_real_fun_real_o(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,R4),S)) ) ) ).

% norm_triangle_mono
tff(fact_5125_norm__triangle__mono,axiom,
    ! [A: complex,R4: real,B: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S)
       => aa_real_o(aa_real_fun_real_o(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,R4),S)) ) ) ).

% norm_triangle_mono
tff(fact_5126_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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_5127_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(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_5128_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => aa_real_o(aa_real_fun_real_o(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_5129_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2) ) ).

% norm_triangle_le
tff(fact_5130_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(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_5131_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(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_5132_norm__power__ineq,axiom,
    ! [X: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(power_power_real(X),Na))),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,X)),Na)) ).

% norm_power_ineq
tff(fact_5133_norm__power__ineq,axiom,
    ! [X: complex,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),Na))),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,X)),Na)) ).

% norm_power_ineq
tff(fact_5134_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(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_5135_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(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_5136_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => aa_real_o(aa_real_fun_real_o(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_5137_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E2) ) ).

% norm_triangle_le_diff
tff(fact_5138_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(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_5139_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(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_5140_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5141_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(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_5142_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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_5143_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(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_5144_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5145_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(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_5146_power__eq__1__iff,axiom,
    ! [W: real,Na: nat] :
      ( ( aa_nat_real(power_power_real(W),Na) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5147_power__eq__1__iff,axiom,
    ! [W: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Na) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5148_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_5149_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_5150_square__norm__one,axiom,
    ! [X: real] :
      ( ( aa_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_5151_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_5152_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(Z),M)),aa_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_5153_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(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_5154_minus__one__div__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ).

% minus_one_div_numeral
tff(fact_5155_one__div__minus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ).

% one_div_minus_numeral
tff(fact_5156_minus__numeral__div__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ).

% minus_numeral_div_numeral
tff(fact_5157_numeral__div__minus__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ).

% numeral_div_minus_numeral
tff(fact_5158_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,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_5159_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(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,R4))
       => 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))),R4)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5160_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5161_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,inc(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_5162_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R4: int] : adjust_div(product_Pair_int_int(Q3,R4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q3),aa_o_int(zero_n2684676970156552555ol_int,R4 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_5163_add__neg__numeral__special_I5_J,axiom,
    ! [Na: 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,numeral_numeral_real(Na))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(Na))) ).

% add_neg_numeral_special(5)
tff(fact_5164_add__neg__numeral__special_I5_J,axiom,
    ! [Na: 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,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(Na))) ).

% add_neg_numeral_special(5)
tff(fact_5165_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(Na))) ).

% add_neg_numeral_special(5)
tff(fact_5166_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(Na))) ).

% add_neg_numeral_special(5)
tff(fact_5167_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(Na))) = uminus_uminus_rat(numeral_numeral_rat(inc(Na))) ).

% add_neg_numeral_special(5)
tff(fact_5168_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5169_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,inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5170_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(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5171_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(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5172_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_5173_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(inc(M)) ).

% diff_numeral_special(6)
tff(fact_5174_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,inc(M)) ).

% diff_numeral_special(6)
tff(fact_5175_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(inc(M)) ).

% diff_numeral_special(6)
tff(fact_5176_diff__numeral__special_I6_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(inc(M)) ).

% diff_numeral_special(6)
tff(fact_5177_diff__numeral__special_I6_J,axiom,
    ! [M: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(inc(M)) ).

% diff_numeral_special(6)
tff(fact_5178_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(Na)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(Na))) ).

% diff_numeral_special(5)
tff(fact_5179_diff__numeral__special_I5_J,axiom,
    ! [Na: 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,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(Na))) ).

% diff_numeral_special(5)
tff(fact_5180_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(Na))) ).

% diff_numeral_special(5)
tff(fact_5181_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(Na))) ).

% diff_numeral_special(5)
tff(fact_5182_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : minus_minus_rat(uminus_uminus_rat(one_one_rat),numeral_numeral_rat(Na)) = uminus_uminus_rat(numeral_numeral_rat(inc(Na))) ).

% diff_numeral_special(5)
tff(fact_5183_num__induct,axiom,
    ! [P: fun_num_o,X: num] :
      ( aa_num_o(P,one)
     => ( ! [X2: num] :
            ( aa_num_o(P,X2)
           => aa_num_o(P,inc(X2)) )
       => aa_num_o(P,X) ) ) ).

% num_induct
tff(fact_5184_add__inc,axiom,
    ! [X: num,Y: num] : plus_plus_num(X,inc(Y)) = inc(plus_plus_num(X,Y)) ).

% add_inc
tff(fact_5185_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R4: int,Q6: int,R3: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q3,R4))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q6,R3))
       => ( Q3 = Q6 ) ) ) ).

% unique_quotient
tff(fact_5186_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R4: int,Q6: int,R3: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q3,R4))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q6,R3))
       => ( R4 = R3 ) ) ) ).

% unique_remainder
tff(fact_5187_inc_Osimps_I1_J,axiom,
    inc(one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_5188_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_5189_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa_num_num(bit1,X)) = aa_num_num(bit0,inc(X)) ).

% inc.simps(3)
tff(fact_5190_add__One,axiom,
    ! [X: num] : plus_plus_num(X,one) = inc(X) ).

% add_One
tff(fact_5191_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_5192_mult__inc,axiom,
    ! [X: num,Y: num] : times_times_num(X,inc(Y)) = plus_plus_num(times_times_num(X,Y),X) ).

% mult_inc
tff(fact_5193_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R4: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q3,R4))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_5194_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R4: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q3,R4))
     => ( aa_int_int(modulo_modulo_int(K),L) = R4 ) ) ).

% mod_int_unique
tff(fact_5195_numeral__inc,axiom,
    ! [X: num] : numera6690914467698888265omplex(inc(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% numeral_inc
tff(fact_5196_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_real(inc(X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(X)),one_one_real) ).

% numeral_inc
tff(fact_5197_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_rat(inc(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ).

% numeral_inc
tff(fact_5198_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,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_5199_numeral__inc,axiom,
    ! [X: num] : aa_num_int(numeral_numeral_int,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_5200_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_5201_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_5202_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q3,R4))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,
            product_Pair_int_int($ite(R4 = 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)),
              $ite(R4 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R4)))) ) ) ).

% zminus1_lemma
tff(fact_5203_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R4: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q3,R4))
    <=> ( ( 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)),R4) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R4),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R4)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R4),zero_zero_int) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5204_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q3,R4))
       => 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))),R4)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5205_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_5206_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5207_product__nth,axiom,
    ! [Na: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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),Na) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_5208_product__nth,axiom,
    ! [Na: nat,Xs: list_Code_integer,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Na) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_5209_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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),Na) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_5210_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Na) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_5211_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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),Na) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_5212_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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),Na) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_5213_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Na) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_5214_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Na) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_5215_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Na) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_5216_product__nth,axiom,
    ! [Na: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Na) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_5217_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_5218_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5219_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5220_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_5221_pochhammer__double,axiom,
    ! [Z: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_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))),Na)) = 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))),Na)))),aa_nat_real(comm_s7457072308508201937r_real(Z),Na))),aa_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),numeral_numeral_real(aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_5222_pochhammer__double,axiom,
    ! [Z: complex,Na: 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))),Na)) = 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))),Na)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Na))),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))))),Na)) ).

% pochhammer_double
tff(fact_5223_pochhammer__double,axiom,
    ! [Z: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_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))),Na)) = 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))),Na)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_5224_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_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_5225_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na)))),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))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_5226_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(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_5227_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5228_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5229_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(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_5230_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_5231_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_5232_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_5233_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_5234_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5235_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_5236_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_5237_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_5238_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_5239_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_5240_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_5241_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_5242_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_5243_dbl__inc__simps_I4_J,axiom,
    neg_nu5219082963157363817nc_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_5244_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_5245_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5246_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5247_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_5248_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_5249_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_5250_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_5251_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_5252_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_5253_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_5254_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_5255_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_5256_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_5257_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_5258_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_5259_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K)) = numeral_numeral_real(bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_5260_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_5261_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,bitM(K)) ).

% dbl_dec_simps(5)
tff(fact_5262_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5263_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5264_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% ln_gt_zero_iff
tff(fact_5265_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_5266_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_5267_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K))) ).

% dbl_dec_simps(1)
tff(fact_5268_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_5269_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_5270_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_5271_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu3179335615603231917ec_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K))) ).

% dbl_dec_simps(1)
tff(fact_5272_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K))) ).

% dbl_inc_simps(1)
tff(fact_5273_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_5274_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_5275_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_5276_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K))) ).

% dbl_inc_simps(1)
tff(fact_5277_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_iff
tff(fact_5278_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_5279_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_5280_ln__less__self,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),X) ) ).

% ln_less_self
tff(fact_5281_pochhammer__pos,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),Na)) ) ).

% pochhammer_pos
tff(fact_5282_pochhammer__pos,axiom,
    ! [X: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),Na)) ) ).

% pochhammer_pos
tff(fact_5283_pochhammer__pos,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),Na)) ) ).

% pochhammer_pos
tff(fact_5284_pochhammer__pos,axiom,
    ! [X: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),Na)) ) ).

% pochhammer_pos
tff(fact_5285_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Na: nat,M: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5286_pochhammer__eq__0__mono,axiom,
    ! [A: real,Na: nat,M: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Na) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5287_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Na: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5288_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,Na: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Na) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5289_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),M) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),Na) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5290_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5291_semiring__norm_I28_J,axiom,
    ! [Na: num] : bitM(aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(bit0,Na)) ).

% semiring_norm(28)
tff(fact_5292_semiring__norm_I27_J,axiom,
    ! [Na: num] : bitM(aa_num_num(bit0,Na)) = aa_num_num(bit1,bitM(Na)) ).

% semiring_norm(27)
tff(fact_5293_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = aa_num_num(bit0,Na) ).

% inc_BitM_eq
tff(fact_5294_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = aa_num_num(bit1,Na) ).

% BitM_inc_eq
tff(fact_5295_ln__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X) ) ).

% ln_bound
tff(fact_5296_ln__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)) ) ).

% ln_gt_zero
tff(fact_5297_ln__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_5298_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5299_ln__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)) ) ).

% ln_ge_zero
tff(fact_5300_pochhammer__nonneg,axiom,
    ! [X: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),Na)) ) ).

% pochhammer_nonneg
tff(fact_5301_pochhammer__nonneg,axiom,
    ! [X: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),Na)) ) ).

% pochhammer_nonneg
tff(fact_5302_pochhammer__nonneg,axiom,
    ! [X: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),Na)) ) ).

% pochhammer_nonneg
tff(fact_5303_pochhammer__nonneg,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),Na)) ) ).

% pochhammer_nonneg
tff(fact_5304_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_5305_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_5306_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_5307_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_5308_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_5309_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bitM(Na))) ).

% eval_nat_numeral(2)
tff(fact_5310_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5311_BitM__plus__one,axiom,
    ! [Na: num] : plus_plus_num(bitM(Na),one) = aa_num_num(bit0,Na) ).

% BitM_plus_one
tff(fact_5312_one__plus__BitM,axiom,
    ! [Na: num] : plus_plus_num(one,bitM(Na)) = aa_num_num(bit0,Na) ).

% one_plus_BitM
tff(fact_5313_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(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_5314_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(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_5315_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5316_ln__div,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(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_5317_pochhammer__rec,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Na)) ).

% pochhammer_rec
tff(fact_5318_pochhammer__rec,axiom,
    ! [A: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Na)) ).

% pochhammer_rec
tff(fact_5319_pochhammer__rec,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),Na)) ).

% pochhammer_rec
tff(fact_5320_pochhammer__rec,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Na)) ).

% pochhammer_rec
tff(fact_5321_pochhammer__rec,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Na)) ).

% pochhammer_rec
tff(fact_5322_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_5323_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_5324_dbl__inc__def,axiom,
    ! [X: rat] : neg_nu5219082963157363817nc_rat(X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_5325_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_5326_pochhammer__Suc,axiom,
    ! [A: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(A),Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Na))) ).

% pochhammer_Suc
tff(fact_5327_pochhammer__Suc,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% pochhammer_Suc
tff(fact_5328_pochhammer__Suc,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Na))) ).

% pochhammer_Suc
tff(fact_5329_pochhammer__Suc,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),Na)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Na))) ).

% pochhammer_Suc
tff(fact_5330_pochhammer__Suc,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Na)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Na))) ).

% pochhammer_Suc
tff(fact_5331_pochhammer__rec_H,axiom,
    ! [Z: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(comm_s7457072308508201937r_real(Z),Na)) ).

% pochhammer_rec'
tff(fact_5332_pochhammer__rec_H,axiom,
    ! [Z: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Na)) = 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,Na))),aa_nat_int(comm_s4660882817536571857er_int(Z),Na)) ).

% pochhammer_rec'
tff(fact_5333_pochhammer__rec_H,axiom,
    ! [Z: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Na)) = 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,Na))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na)) ).

% pochhammer_rec'
tff(fact_5334_pochhammer__rec_H,axiom,
    ! [Z: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Na))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Na)) ).

% pochhammer_rec'
tff(fact_5335_pochhammer__rec_H,axiom,
    ! [Z: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Na))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na)) ).

% pochhammer_rec'
tff(fact_5336_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5337_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5338_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5339_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5340_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5341_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5342_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5343_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5344_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),K) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5345_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_5346_pochhammer__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),Na) = zero_zero_real )
    <=> ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5347_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = zero_zero_complex )
    <=> ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5348_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = zero_zero_rat )
    <=> ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
          & ( A = uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5349_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5350_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5351_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5352_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5353_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5354_pochhammer__product_H,axiom,
    ! [Z: real,Na: nat,M: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),Na)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Na))),M)) ).

% pochhammer_product'
tff(fact_5355_pochhammer__product_H,axiom,
    ! [Z: int,Na: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Na)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Na))),M)) ).

% pochhammer_product'
tff(fact_5356_pochhammer__product_H,axiom,
    ! [Z: nat,Na: 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,Na),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na)),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,Na))),M)) ).

% pochhammer_product'
tff(fact_5357_pochhammer__product_H,axiom,
    ! [Z: complex,Na: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Na)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Na))),M)) ).

% pochhammer_product'
tff(fact_5358_pochhammer__product_H,axiom,
    ! [Z: rat,Na: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Na))),M)) ).

% pochhammer_product'
tff(fact_5359_binomial__mono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(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)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)) ) ) ).

% binomial_mono
tff(fact_5360_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(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))),Na)),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))),Na)),Na)) ).

% binomial_maximum'
tff(fact_5361_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_5362_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_5363_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_5364_numeral__BitM,axiom,
    ! [Na: num] : numera6690914467698888265omplex(bitM(Na)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Na))),one_one_complex) ).

% numeral_BitM
tff(fact_5365_numeral__BitM,axiom,
    ! [Na: num] : numeral_numeral_real(bitM(Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(aa_num_num(bit0,Na))),one_one_real) ).

% numeral_BitM
tff(fact_5366_numeral__BitM,axiom,
    ! [Na: num] : numeral_numeral_rat(bitM(Na)) = minus_minus_rat(numeral_numeral_rat(aa_num_num(bit0,Na)),one_one_rat) ).

% numeral_BitM
tff(fact_5367_numeral__BitM,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,bitM(Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),one_one_int) ).

% numeral_BitM
tff(fact_5368_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(bitM(W))) ).

% odd_numeral_BitM
tff(fact_5369_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,bitM(W))) ).

% odd_numeral_BitM
tff(fact_5370_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,bitM(W))) ).

% odd_numeral_BitM
tff(fact_5371_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(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_5372_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(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_5373_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(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_5374_ln__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_nat_real(power_power_real(X),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_5375_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),M)),aa_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,Na),M))) ) ) ).

% pochhammer_product
tff(fact_5376_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Na) = 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,Na),M))) ) ) ).

% pochhammer_product
tff(fact_5377_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na) = 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,Na),M))) ) ) ).

% pochhammer_product
tff(fact_5378_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),Na) = 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,Na),M))) ) ) ).

% pochhammer_product
tff(fact_5379_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% pochhammer_product
tff(fact_5380_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(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)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)) ) ) ).

% binomial_strict_mono
tff(fact_5381_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_5382_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_5383_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_5384_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(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_5385_pochhammer__absorb__comp,axiom,
    ! [R4: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R4,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R4)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R4),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R4),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_5386_pochhammer__absorb__comp,axiom,
    ! [R4: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R4),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R4)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R4)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_5387_pochhammer__absorb__comp,axiom,
    ! [R4: 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,R4),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R4)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R4),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R4)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_5388_pochhammer__absorb__comp,axiom,
    ! [R4: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R4),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R4)),K)) = aa_complex_complex(times_times_complex(R4),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R4)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_5389_pochhammer__absorb__comp,axiom,
    ! [R4: rat,K: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(R4,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(R4)),K)) = aa_rat_rat(times_times_rat(R4),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(R4)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_5390_ln__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_5391_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_5392_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_5393_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_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(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_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_5394_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_5395_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_5396_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_5397_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_5398_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_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(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_5399_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_5400_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_5401_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_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(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K)) ).

% pochhammer_minus'
tff(fact_5402_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_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_5403_arcosh__real__def,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_5404_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_5405_choose__two,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_5406_binomial__n__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_5407_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),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,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_5408_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_5409_binomial__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,zero_zero_nat)) = Na ).

% binomial_1
tff(fact_5410_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_5411_binomial__Suc__n,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),Na) = aa_nat_nat(suc,Na) ).

% binomial_Suc_n
tff(fact_5412_binomial__n__n,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),Na) = one_one_nat ).

% binomial_n_n
tff(fact_5413_choose__one,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),one_one_nat) = Na ).

% choose_one
tff(fact_5414_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_5415_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_5416_Suc__times__binomial,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ).

% Suc_times_binomial
tff(fact_5417_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_5418_choose__mult__lemma,axiom,
    ! [M: nat,R4: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R4)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R4)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R4)),M)) ).

% choose_mult_lemma
tff(fact_5419_binomial__le__pow,axiom,
    ! [R4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R4),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),R4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R4)) ) ).

% binomial_le_pow
tff(fact_5420_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ).

% zero_less_binomial
tff(fact_5421_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_5422_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_5423_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_5424_binomial__absorb__comp,axiom,
    ! [Na: 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,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_5425_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_5426_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Na),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_5427_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_5428_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% binomial_le_pow2
tff(fact_5429_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),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,Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_5430_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_5431_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),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,Na),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,Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_5432_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_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_5433_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5434_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_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_5435_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_5436_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5437_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X22: num] : size_option_num(X,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5438_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_5439_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_5440_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_5441_abs__abs,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_abs
tff(fact_5442_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_5443_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_5444_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5445_abs__idempotent,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_idempotent
tff(fact_5446_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5447_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_5448_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_5449_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_5450_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5451_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_5452_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_5453_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_5454_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5455_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5456_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5457_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5458_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5459_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_5460_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5461_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5462_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5463_abs__numeral,axiom,
    ! [Na: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Na)) = numera6620942414471956472nteger(Na) ).

% abs_numeral
tff(fact_5464_abs__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,numeral_numeral_real(Na)) = numeral_numeral_real(Na) ).

% abs_numeral
tff(fact_5465_abs__numeral,axiom,
    ! [Na: num] : abs_abs_rat(numeral_numeral_rat(Na)) = numeral_numeral_rat(Na) ).

% abs_numeral
tff(fact_5466_abs__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% abs_numeral
tff(fact_5467_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_5468_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_5469_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_5470_abs__mult__self__eq,axiom,
    ! [A: 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,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_5471_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_5472_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_5473_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_5474_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_5475_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_5476_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_5477_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_5478_abs__add__abs,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_add_abs
tff(fact_5479_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_5480_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_5481_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_5482_abs__divide,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ).

% abs_divide
tff(fact_5483_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_5484_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_5485_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_5486_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_5487_abs__minus,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus
tff(fact_5488_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_5489_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_5490_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_5491_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_5492_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% abs_dvd_iff
tff(fact_5493_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% abs_dvd_iff
tff(fact_5494_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% abs_dvd_iff
tff(fact_5495_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,abs_abs_rat(M)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% abs_dvd_iff
tff(fact_5496_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% dvd_abs_iff
tff(fact_5497_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% dvd_abs_iff
tff(fact_5498_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% dvd_abs_iff
tff(fact_5499_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),abs_abs_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% dvd_abs_iff
tff(fact_5500_abs__of__nat,axiom,
    ! [Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) ).

% abs_of_nat
tff(fact_5501_abs__of__nat,axiom,
    ! [Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% abs_of_nat
tff(fact_5502_abs__of__nat,axiom,
    ! [Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% abs_of_nat
tff(fact_5503_abs__of__nat,axiom,
    ! [Na: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% abs_of_nat
tff(fact_5504_abs__bool__eq,axiom,
    ! [P: $o] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real((P))) = zero_n3304061248610475627l_real((P)) ).

% abs_bool_eq
tff(fact_5505_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_rat(zero_n2052037380579107095ol_rat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% abs_bool_eq
tff(fact_5506_abs__bool__eq,axiom,
    ! [P: $o] : aa_int_int(abs_abs_int,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% abs_bool_eq
tff(fact_5507_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(aa_o_Code_integer(zero_n356916108424825756nteger,(P))) = aa_o_Code_integer(zero_n356916108424825756nteger,(P)) ).

% abs_bool_eq
tff(fact_5508_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_5509_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5510_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% tanh_real_less_iff
tff(fact_5511_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_5512_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_5513_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5514_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5515_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_5516_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_5517_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_5518_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_5519_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_5520_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_5521_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_5522_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_5523_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_5524_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_5525_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),abs_abs_rat(A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_5526_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_5527_abs__neg__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = numeral_numeral_real(Na) ).

% abs_neg_numeral
tff(fact_5528_abs__neg__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,Na) ).

% abs_neg_numeral
tff(fact_5529_abs__neg__numeral,axiom,
    ! [Na: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(Na) ).

% abs_neg_numeral
tff(fact_5530_abs__neg__numeral,axiom,
    ! [Na: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(Na))) = numeral_numeral_rat(Na) ).

% abs_neg_numeral
tff(fact_5531_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_5532_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_5533_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_5534_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_5535_abs__power__minus,axiom,
    ! [A: real,Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),Na)) = aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),Na)) ).

% abs_power_minus
tff(fact_5536_abs__power__minus,axiom,
    ! [A: int,Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Na)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Na)) ).

% abs_power_minus
tff(fact_5537_abs__power__minus,axiom,
    ! [A: code_integer,Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Na)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) ).

% abs_power_minus
tff(fact_5538_abs__power__minus,axiom,
    ! [A: rat,Na: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Na)) = abs_abs_rat(aa_nat_rat(power_power_rat(A),Na)) ).

% abs_power_minus
tff(fact_5539_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% zero_less_arctan_iff
tff(fact_5540_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_5541_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_5542_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_5543_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% tanh_real_pos_iff
tff(fact_5544_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_5545_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,abs_abs_rat(B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5546_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5547_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,abs_abs_rat(B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5548_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5549_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_5550_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_nonpos
tff(fact_5551_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_5552_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5553_artanh__minus__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5554_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5555_zero__less__power__abs__iff,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Na))
    <=> ( ( A != zero_zero_real )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5556_zero__less__power__abs__iff,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),Na))
    <=> ( ( A != zero_zero_rat )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5557_zero__less__power__abs__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na))
    <=> ( ( A != zero_zero_int )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5558_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_5559_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_5560_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5561_abs__power2,axiom,
    ! [A: 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_5562_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_5563_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_5564_power2__abs,axiom,
    ! [A: real] : aa_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5565_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_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_5566_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_5567_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_5568_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5569_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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(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_5570_real__sqrt__abs,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_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_5571_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_5572_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_5573_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_5574_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_5575_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_5576_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),abs_abs_rat(A)) ).

% abs_ge_self
tff(fact_5577_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_5578_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_5579_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5580_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_5581_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_5582_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_5583_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_5584_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : 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_mult
tff(fact_5585_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_5586_abs__mult,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_mult
tff(fact_5587_abs__mult,axiom,
    ! [A: int,B: 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_mult
tff(fact_5588_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5589_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5590_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5591_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5592_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_5593_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_5594_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(minus_minus_rat(A,B)) = abs_abs_rat(minus_minus_rat(B,A)) ).

% abs_minus_commute
tff(fact_5595_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_5596_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5597_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5598_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5599_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( abs_abs_rat(X) = abs_abs_rat(Y) )
    <=> ( ( X = Y )
        | ( X = uminus_uminus_rat(Y) ) ) ) ).

% abs_eq_iff
tff(fact_5600_power__abs,axiom,
    ! [A: code_integer,Na: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na) ).

% power_abs
tff(fact_5601_power__abs,axiom,
    ! [A: int,Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Na)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na) ).

% power_abs
tff(fact_5602_power__abs,axiom,
    ! [A: real,Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(power_power_real(A),Na)) = aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Na) ).

% power_abs
tff(fact_5603_power__abs,axiom,
    ! [A: rat,Na: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),Na)) = aa_nat_rat(power_power_rat(abs_abs_rat(A)),Na) ).

% power_abs
tff(fact_5604_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5605_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5606_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5607_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( abs_abs_rat(L) = abs_abs_rat(K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5608_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_5609_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% arctan_less_iff
tff(fact_5610_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_5611_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A)) ).

% abs_ge_zero
tff(fact_5612_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_5613_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_5614_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_5615_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_5616_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_5617_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_5618_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_5619_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_5620_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_5621_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_5622_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5623_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq
tff(fact_5624_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_5625_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5626_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(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),D)) ) ) ).

% abs_mult_less
tff(fact_5627_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_5628_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_5629_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(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),D)) ) ) ).

% abs_mult_less
tff(fact_5630_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5631_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5632_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_5633_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5634_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5635_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B)))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq3
tff(fact_5636_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_5637_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5638_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5639_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq2
tff(fact_5640_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_5641_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5642_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_5643_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).

% nonzero_abs_divide
tff(fact_5644_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_5645_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B) ) ) ).

% abs_leI
tff(fact_5646_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_5647_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_5648_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_5649_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B) ) ).

% abs_le_D2
tff(fact_5650_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_5651_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_5652_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_5653_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B) ) ) ).

% abs_le_iff
tff(fact_5654_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_5655_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_5656_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_5657_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),abs_abs_rat(A)) ).

% abs_ge_minus_self
tff(fact_5658_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_5659_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_5660_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_5661_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_5662_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_5663_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),B) ) ) ).

% abs_less_iff
tff(fact_5664_tanh__real__lt__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),one_one_real) ).

% tanh_real_lt_1
tff(fact_5665_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(X)),E) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5666_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5667_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5668_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_rat(times_times_rat(abs_abs_rat(Y)),X) = abs_abs_rat(aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5669_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(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_5670_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5671_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5672_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_5673_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(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_5674_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(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_5675_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_5676_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_5677_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(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_5678_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(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_5679_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5680_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat(A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = uminus_uminus_rat(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5681_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5682_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5683_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5684_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = abs_abs_rat(B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = uminus_uminus_rat(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5685_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5686_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5687_zero__le__power__abs,axiom,
    ! [A: code_integer,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na)) ).

% zero_le_power_abs
tff(fact_5688_zero__le__power__abs,axiom,
    ! [A: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),Na)) ).

% zero_le_power_abs
tff(fact_5689_zero__le__power__abs,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na)) ).

% zero_le_power_abs
tff(fact_5690_zero__le__power__abs,axiom,
    ! [A: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Na)) ).

% zero_le_power_abs
tff(fact_5691_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5692_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( divide_divide_rat(abs_abs_rat(X),Y) = abs_abs_rat(divide_divide_rat(X,Y)) ) ) ).

% abs_div_pos
tff(fact_5693_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_5694_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(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_5695_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_5696_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_neg
tff(fact_5697_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_5698_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_5699_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_5700_abs__if,axiom,
    ! [A: rat] :
      abs_abs_rat(A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),uminus_uminus_rat(A),A) ).

% abs_if
tff(fact_5701_abs__if__raw,axiom,
    ! [X3: real] :
      aa_real_real(abs_abs_real,X3) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real),aa_real_real(uminus_uminus_real,X3),X3) ).

% abs_if_raw
tff(fact_5702_abs__if__raw,axiom,
    ! [X3: int] :
      aa_int_int(abs_abs_int,X3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int),aa_int_int(uminus_uminus_int,X3),X3) ).

% abs_if_raw
tff(fact_5703_abs__if__raw,axiom,
    ! [X3: code_integer] :
      abs_abs_Code_integer(X3) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3),X3) ).

% abs_if_raw
tff(fact_5704_abs__if__raw,axiom,
    ! [X3: rat] :
      abs_abs_rat(X3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat),uminus_uminus_rat(X3),X3) ).

% abs_if_raw
tff(fact_5705_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5706_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A,B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq4
tff(fact_5707_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(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_5708_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(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_5709_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_5710_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(minus_minus_rat(A,C))),abs_abs_rat(minus_minus_rat(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_5711_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(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),D)))),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),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5712_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5713_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R4)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R4)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R4)) ) ) ).

% abs_diff_le_iff
tff(fact_5714_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(X,A))),R4)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,R4)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R4)) ) ) ).

% abs_diff_le_iff
tff(fact_5715_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R4)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R4)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R4)) ) ) ).

% abs_diff_le_iff
tff(fact_5716_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R4)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R4)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R4)) ) ) ).

% abs_diff_le_iff
tff(fact_5717_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R4)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R4)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R4)) ) ) ).

% abs_diff_less_iff
tff(fact_5718_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R4)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R4)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R4)) ) ) ).

% abs_diff_less_iff
tff(fact_5719_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(X,A))),R4)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,R4)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R4)) ) ) ).

% abs_diff_less_iff
tff(fact_5720_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R4)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R4)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R4)) ) ) ).

% abs_diff_less_iff
tff(fact_5721_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_5722_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_5723_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(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_5724_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(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_5725_tanh__real__gt__neg1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X)) ).

% tanh_real_gt_neg1
tff(fact_5726_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X))) ).

% abs_add_one_gt_zero
tff(fact_5727_abs__add__one__gt__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_5728_abs__add__one__gt__zero,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),abs_abs_rat(X))) ).

% abs_add_one_gt_zero
tff(fact_5729_abs__add__one__gt__zero,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(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_5730_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_5731_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5732_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(X)),abs_abs_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(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_5733_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(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_5734_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5735_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_5736_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_5737_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( aa_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_5738_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 )
    <=> ( abs_abs_rat(X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_5739_power__even__abs,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Na) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Na) ) ) ).

% power_even_abs
tff(fact_5740_power__even__abs,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Na) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_even_abs
tff(fact_5741_power__even__abs,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Na) = aa_nat_real(power_power_real(A),Na) ) ) ).

% power_even_abs
tff(fact_5742_power__even__abs,axiom,
    ! [Na: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),Na) = aa_nat_rat(power_power_rat(A),Na) ) ) ).

% power_even_abs
tff(fact_5743_arctan__double,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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,numeral_numeral_real(aa_num_num(bit0,one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_5744_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5745_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(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))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5746_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(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))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5747_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5748_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_5749_abs__square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_5750_abs__square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(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)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_5751_abs__square__le__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real) ) ).

% abs_square_le_1
tff(fact_5752_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_5753_abs__square__less__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real) ) ).

% abs_square_less_1
tff(fact_5754_abs__square__less__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(X)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_5755_abs__square__less__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_less_1
tff(fact_5756_power__mono__even,axiom,
    ! [Na: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Na)) ) ) ).

% power_mono_even
tff(fact_5757_power__mono__even,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),abs_abs_rat(B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono_even
tff(fact_5758_power__mono__even,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono_even
tff(fact_5759_power__mono__even,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(A),Na)),aa_nat_real(power_power_real(B),Na)) ) ) ).

% power_mono_even
tff(fact_5760_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_5761_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_5762_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_5763_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_5764_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5765_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5766_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_5767_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_5768_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(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(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_5769_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,X: code_integer] :
      ( ! [X2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_5770_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,X: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X2)
         => aa_rat_o(aa_rat_fun_rat_o(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,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_5771_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,X: int] :
      ( ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X2)
         => aa_int_o(aa_int_fun_int_o(P,X2),aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(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_5772_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,X: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
         => aa_real_o(aa_real_fun_real_o(P,X2),aa_nat_real(power_power_real(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,X)),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5773_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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,Na))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5774_fact__double,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(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))),Na))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Na))),aa_nat_complex(semiri5044797733671781792omplex,Na)) ).

% fact_double
tff(fact_5775_fact__double,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_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))),Na))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),Na))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_double
tff(fact_5776_fact__double,axiom,
    ! [Na: nat] : 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))),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_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))),Na))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),Na))),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_double
tff(fact_5777_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: 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,Na))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,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,M),Na))),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,Na),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))))) ) ).

% modulo_int_unfold
tff(fact_5778_mask__numeral,axiom,
    ! [Na: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Na)) = 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(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,Na)))) ).

% mask_numeral
tff(fact_5779_mask__numeral,axiom,
    ! [Na: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Na)) = 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_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,Na)))) ).

% mask_numeral
tff(fact_5780_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_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,Na))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),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),Na)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_5781_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_5782_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_5783_sgn__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(sgn_sgn_complex,A)) = aa_complex_complex(sgn_sgn_complex,A) ).

% sgn_sgn
tff(fact_5784_sgn__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ).

% sgn_sgn
tff(fact_5785_sgn__sgn,axiom,
    ! [A: rat] : sgn_sgn_rat(sgn_sgn_rat(A)) = sgn_sgn_rat(A) ).

% sgn_sgn
tff(fact_5786_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_5787_sgn__0,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_5788_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_5789_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_5790_sgn__0,axiom,
    sgn_sgn_rat(zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_5791_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_5792_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_5793_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_5794_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_5795_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_5796_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_5797_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_5798_sgn__1,axiom,
    sgn_sgn_rat(one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_5799_sgn__one,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_one
tff(fact_5800_sgn__one,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_one
tff(fact_5801_sgn__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_divide
tff(fact_5802_sgn__divide,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_divide
tff(fact_5803_sgn__divide,axiom,
    ! [A: rat,B: rat] : sgn_sgn_rat(divide_divide_rat(A,B)) = divide_divide_rat(sgn_sgn_rat(A),sgn_sgn_rat(B)) ).

% sgn_divide
tff(fact_5804_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5805_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5806_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5807_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5808_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] : sgn_sgn_rat(uminus_uminus_rat(A)) = uminus_uminus_rat(sgn_sgn_rat(A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5809_power__sgn,axiom,
    ! [A: code_integer,Na: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Na)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),Na) ).

% power_sgn
tff(fact_5810_power__sgn,axiom,
    ! [A: int,Na: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(power_power_int(A),Na)) = aa_nat_int(power_power_int(aa_int_int(sgn_sgn_int,A)),Na) ).

% power_sgn
tff(fact_5811_power__sgn,axiom,
    ! [A: real,Na: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(power_power_real(A),Na)) = aa_nat_real(power_power_real(aa_real_real(sgn_sgn_real,A)),Na) ).

% power_sgn
tff(fact_5812_power__sgn,axiom,
    ! [A: rat,Na: nat] : sgn_sgn_rat(aa_nat_rat(power_power_rat(A),Na)) = aa_nat_rat(power_power_rat(sgn_sgn_rat(A)),Na) ).

% power_sgn
tff(fact_5813_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_5814_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_5815_sgn__greater,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_greater
tff(fact_5816_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),sgn_sgn_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_5817_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_5818_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_5819_sgn__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_less
tff(fact_5820_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,sgn_sgn_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_less
tff(fact_5821_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_5822_divide__sgn,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(sgn_sgn_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(sgn_sgn_real,B)) ).

% divide_sgn
tff(fact_5823_divide__sgn,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,sgn_sgn_rat(B)) = aa_rat_rat(times_times_rat(A),sgn_sgn_rat(B)) ).

% divide_sgn
tff(fact_5824_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_5825_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_5826_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_5827_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_5828_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_5829_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_5830_mask__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = zero_zero_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_5831_mask__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,Na) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_5832_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_5833_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_5834_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_5835_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_5836_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_5837_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_5838_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_5839_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_5840_sgn__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_5841_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( sgn_sgn_rat(A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_5842_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_5843_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,M)),Na) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5844_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),Na) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5845_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_5846_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_5847_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(sgn_sgn_rat(A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_5848_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_5849_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_5850_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_5851_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_5852_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_5853_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_5854_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,M)),Na) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5855_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),Na) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5856_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_Suc
tff(fact_5857_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Na))),aa_nat_complex(semiri5044797733671781792omplex,Na)) ).

% fact_Suc
tff(fact_5858_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_Suc
tff(fact_5859_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_Suc
tff(fact_5860_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_Suc
tff(fact_5861_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% sgn_mult_self_eq
tff(fact_5862_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(sgn_sgn_rat(A)),sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_5863_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_5864_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_5865_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_5866_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_5867_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(A != zero_zero_complex) ).

% sgn_abs
tff(fact_5868_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% sgn_abs
tff(fact_5869_sgn__abs,axiom,
    ! [A: rat] : abs_abs_rat(sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_abs
tff(fact_5870_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_abs
tff(fact_5871_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).

% sgn_abs
tff(fact_5872_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(A != zero_zero_complex) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5873_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5874_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : sgn_sgn_rat(abs_abs_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5875_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5876_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,abs_abs_Code_integer(A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5877_take__bit__minus__one__eq__mask,axiom,
    ! [Na: nat] : bit_se1745604003318907178nteger(Na,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,Na) ).

% take_bit_minus_one_eq_mask
tff(fact_5878_take__bit__minus__one__eq__mask,axiom,
    ! [Na: nat] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,Na) ).

% take_bit_minus_one_eq_mask
tff(fact_5879_sgn__mult__dvd__iff,axiom,
    ! [R4: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R4)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R4 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_5880_mult__sgn__dvd__iff,axiom,
    ! [L: int,R4: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R4))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R4 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_5881_dvd__sgn__mult__iff,axiom,
    ! [L: int,R4: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R4)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R4 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_5882_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R4)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R4 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_5883_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_5884_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_5885_sgn__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_5886_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_5887_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_5888_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) ) ) ).

% sgn_neg
tff(fact_5889_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_5890_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_5891_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_5892_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_5893_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_5894_sgn__of__nat,axiom,
    ! [Na: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = zero_n3304061248610475627l_real(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_5895_sgn__of__nat,axiom,
    ! [Na: nat] : sgn_sgn_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)) = zero_n2052037380579107095ol_rat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_5896_sgn__of__nat,axiom,
    ! [Na: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_5897_sgn__of__nat,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% sgn_of_nat
tff(fact_5898_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Na)) ) ).

% bit_numeral_simps(2)
tff(fact_5899_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Na)) ) ).

% bit_numeral_simps(2)
tff(fact_5900_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_5901_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Na)) ) ).

% bit_numeral_simps(3)
tff(fact_5902_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Na)) ) ).

% bit_numeral_simps(3)
tff(fact_5903_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_5904_bit__0,axiom,
    ! [A: code_integer] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),zero_zero_nat)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_5905_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),zero_zero_nat)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_5906_bit__0,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),zero_zero_nat)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_5907_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(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,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Na)) ) ).

% bit_minus_numeral_int(1)
tff(fact_5908_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(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,Na))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Na)) ) ).

% bit_minus_numeral_int(2)
tff(fact_5909_bit__mod__2__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5910_bit__mod__2__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5911_bit__mod__2__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
    <=> ( ( Na = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5912_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_5913_of__nat__mask__eq,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Na) ).

% of_nat_mask_eq
tff(fact_5914_of__nat__mask__eq,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_int(bit_se2000444600071755411sk_int,Na) ).

% of_nat_mask_eq
tff(fact_5915_bit__of__nat__iff__bit,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_nat_int(semiri1314217659103216013at_int,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Na) ) ).

% bit_of_nat_iff_bit
tff(fact_5916_bit__of__nat__iff__bit,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Na) ) ).

% bit_of_nat_iff_bit
tff(fact_5917_bit__numeral__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),Na) ) ).

% bit_numeral_iff
tff(fact_5918_bit__numeral__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,M)),Na) ) ).

% bit_numeral_iff
tff(fact_5919_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Na)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
          | aa_nat_o(bit_se1146084159140164899it_int(B),Na) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5920_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Na) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5921_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono_nat
tff(fact_5922_fact__ge__self,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_self
tff(fact_5923_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_5924_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_5925_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_5926_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_5927_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_5928_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_5929_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_5930_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_5931_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_5932_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X) = zero_zero_complex )
    <=> ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_5933_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sgn_sgn_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_5934_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_5935_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).

% sgn_mult
tff(fact_5936_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_5937_sgn__mult,axiom,
    ! [A: rat,B: rat] : sgn_sgn_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(sgn_sgn_rat(A)),sgn_sgn_rat(B)) ).

% sgn_mult
tff(fact_5938_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_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(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_5939_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,plus_p5714425477246183910nteger(A,B)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5940_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5941_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) = sgn_sgn_rat(A) )
     => ( sgn_sgn_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = sgn_sgn_rat(A) ) ) ).

% same_sgn_sgn_add
tff(fact_5942_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5943_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,Na) != zero_zero_complex ).

% fact_nonzero
tff(fact_5944_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) != zero_zero_rat ).

% fact_nonzero
tff(fact_5945_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) != zero_zero_int ).

% fact_nonzero
tff(fact_5946_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) != zero_zero_nat ).

% fact_nonzero
tff(fact_5947_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) != zero_zero_real ).

% fact_nonzero
tff(fact_5948_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_5949_bit__unset__bit__iff,axiom,
    ! [M: nat,A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)),Na)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
        & ( M != Na ) ) ) ).

% bit_unset_bit_iff
tff(fact_5950_bit__unset__bit__iff,axiom,
    ! [M: nat,A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
        & ( M != Na ) ) ) ).

% bit_unset_bit_iff
tff(fact_5951_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_5952_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(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(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_5953_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = 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(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_5954_not__bit__1__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,Na)) ).

% not_bit_1_Suc
tff(fact_5955_not__bit__1__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Na)) ).

% not_bit_1_Suc
tff(fact_5956_bit__1__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5957_bit__1__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5958_bit__numeral__simps_I1_J,axiom,
    ! [Na: num] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_num_nat(numeral_numeral_nat,Na)) ).

% bit_numeral_simps(1)
tff(fact_5959_bit__numeral__simps_I1_J,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% bit_numeral_simps(1)
tff(fact_5960_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_5961_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(M),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        & aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ) ).

% bit_take_bit_iff
tff(fact_5962_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ) ).

% bit_take_bit_iff
tff(fact_5963_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5964_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5965_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5966_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) != sgn_sgn_rat(A) )
     => ( ( sgn_sgn_rat(A) != zero_zero_rat )
       => ( ( sgn_sgn_rat(B) != zero_zero_rat )
         => ( sgn_sgn_rat(A) = uminus_uminus_rat(sgn_sgn_rat(B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5967_bit__of__bool__iff,axiom,
    ! [B: $o,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(aa_o_Code_integer(zero_n356916108424825756nteger,(B))),Na)
    <=> ( (B)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5968_bit__of__bool__iff,axiom,
    ! [B: $o,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),Na)
    <=> ( (B)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5969_bit__of__bool__iff,axiom,
    ! [B: $o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),Na)
    <=> ( (B)
        & ( Na = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5970_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_5971_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_5972_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_5973_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_5974_sgn__minus__1,axiom,
    sgn_sgn_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% sgn_minus_1
tff(fact_5975_fact__ge__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_zero
tff(fact_5976_fact__ge__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_zero
tff(fact_5977_fact__ge__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_zero
tff(fact_5978_fact__ge__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_zero
tff(fact_5979_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5980_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5981_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : abs_abs_rat(K) = aa_rat_rat(times_times_rat(K),sgn_sgn_rat(K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5982_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5983_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_5984_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_5985_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_5986_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),sgn_sgn_rat(A)) = A ).

% abs_mult_sgn
tff(fact_5987_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_5988_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_5989_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_5990_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_5991_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(sgn_sgn_rat(A)),abs_abs_rat(A)) = A ).

% sgn_mult_abs
tff(fact_5992_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_5993_mult__sgn__abs,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)),abs_abs_Code_integer(X)) = X ).

% mult_sgn_abs
tff(fact_5994_mult__sgn__abs,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),aa_real_real(abs_abs_real,X)) = X ).

% mult_sgn_abs
tff(fact_5995_mult__sgn__abs,axiom,
    ! [X: rat] : aa_rat_rat(times_times_rat(sgn_sgn_rat(X)),abs_abs_rat(X)) = X ).

% mult_sgn_abs
tff(fact_5996_mult__sgn__abs,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ).

% mult_sgn_abs
tff(fact_5997_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),zero_zero_rat) ).

% fact_not_neg
tff(fact_5998_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),zero_zero_int) ).

% fact_not_neg
tff(fact_5999_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),zero_zero_nat) ).

% fact_not_neg
tff(fact_6000_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Na)),zero_zero_real) ).

% fact_not_neg
tff(fact_6001_fact__gt__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_gt_zero
tff(fact_6002_fact__gt__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_gt_zero
tff(fact_6003_fact__gt__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_gt_zero
tff(fact_6004_fact__gt__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_gt_zero
tff(fact_6005_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_6006_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ) ) ).

% same_sgn_abs_add
tff(fact_6007_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( 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)) ) ) ).

% same_sgn_abs_add
tff(fact_6008_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) = sgn_sgn_rat(A) )
     => ( abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% same_sgn_abs_add
tff(fact_6009_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( 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)) ) ) ).

% same_sgn_abs_add
tff(fact_6010_fact__ge__1,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_1
tff(fact_6011_fact__ge__1,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_1
tff(fact_6012_fact__ge__1,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_1
tff(fact_6013_fact__ge__1,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_1
tff(fact_6014_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% fact_mono
tff(fact_6015_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% fact_mono
tff(fact_6016_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono
tff(fact_6017_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% fact_mono
tff(fact_6018_signed__take__bit__eq__if__positive,axiom,
    ! [A: int,Na: nat] :
      ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),Na)
     => ( aa_int_int(bit_ri631733984087533419it_int(Na),A) = aa_int_int(bit_se2923211474154528505it_int(Na),A) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_6019_abs__zmult__eq__1,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6020_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,M)) ) ).

% fact_dvd
tff(fact_6021_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) ) ).

% fact_dvd
tff(fact_6022_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) ) ).

% fact_dvd
tff(fact_6023_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,M)) ) ).

% fact_dvd
tff(fact_6024_abs__div,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),X)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_6025_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,Na) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Na) ).

% pochhammer_fact
tff(fact_6026_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Na) ).

% pochhammer_fact
tff(fact_6027_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),Na) ).

% pochhammer_fact
tff(fact_6028_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Na) ).

% pochhammer_fact
tff(fact_6029_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) = aa_nat_real(comm_s7457072308508201937r_real(one_one_real),Na) ).

% pochhammer_fact
tff(fact_6030_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% mask_nonnegative_int
tff(fact_6031_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_6032_eucl__rel__int__remainderI,axiom,
    ! [R4: int,L: int,K: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R4)),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)),R4) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q3,R4)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_6033_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_6034_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_6035_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_1_pos
tff(fact_6036_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = one_one_rat )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_6037_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_6038_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_6039_abs__sgn__eq,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% abs_sgn_eq
tff(fact_6040_abs__sgn__eq,axiom,
    ! [A: rat] :
      abs_abs_rat(sgn_sgn_rat(A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_6041_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_6042_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% dvd_fact
tff(fact_6043_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) ) )
       => ( ! [Q2: int] :
              ( ( A32 = product_Pair_int_int(Q2,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
         => ~ ! [R2: int,Q2: int] :
                ( ( A32 = product_Pair_int_int(Q2,R2) )
               => ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),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,Q2),A22)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_6044_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K2) ) )
        | ? [L3: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L3) ) )
        | ? [R5: int,L3: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q4,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3))
            & ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_6045_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_6046_bit__not__int__iff_H,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),Na)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% bit_not_int_iff'
tff(fact_6047_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ) ).

% fact_less_mono
tff(fact_6048_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ) ).

% fact_less_mono
tff(fact_6049_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono
tff(fact_6050_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ) ).

% fact_less_mono
tff(fact_6051_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(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_6052_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Na))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_6053_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Na))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_6054_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,Na))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_6055_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Na))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_6056_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_6057_fact__mod,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_6058_fact__mod,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Na),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_6059_fact__mod,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_6060_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_6061_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_6062_fact__le__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_6063_fact__le__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_6064_fact__le__power,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_6065_fact__le__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_6066_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(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_6067_flip__bit__eq__if,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_se2159334234014336723it_int(Na),A) = aa_int_int(aa_nat_fun_int_int($ite(aa_nat_o(bit_se1146084159140164899it_int(A),Na),bit_se4203085406695923979it_int,bit_se7879613467334960850it_int),Na),A) ).

% flip_bit_eq_if
tff(fact_6068_flip__bit__eq__if,axiom,
    ! [Na: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Na),A) = aa_nat_nat(aa_nat_fun_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na),bit_se4205575877204974255it_nat,bit_se7882103937844011126it_nat),Na),A) ).

% flip_bit_eq_if
tff(fact_6069_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_6070_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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)),Na)) = 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)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_6071_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_1_neg
tff(fact_6072_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_6073_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_6074_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_6075_sgn__if,axiom,
    ! [X: real] :
      aa_real_real(sgn_sgn_real,X) = $ite(
        X = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_if
tff(fact_6076_sgn__if,axiom,
    ! [X: int] :
      aa_int_int(sgn_sgn_int,X) = $ite(
        X = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_6077_sgn__if,axiom,
    ! [X: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = $ite(
        X = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_6078_sgn__if,axiom,
    ! [X: rat] :
      sgn_sgn_rat(X) = $ite(
        X = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_if
tff(fact_6079_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_6080_fact__div__fact__le__pow,axiom,
    ! [R4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R4),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),R4)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R4)) ) ).

% fact_div_fact_le_pow
tff(fact_6081_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_6082_norm__sgn,axiom,
    ! [X: real] :
      aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = $ite(X = zero_zero_real,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_6083_norm__sgn,axiom,
    ! [X: complex] :
      aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = $ite(X = zero_zero_complex,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_6084_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6085_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(M,K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6086_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Na)) ) ).

% choose_dvd
tff(fact_6087_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% choose_dvd
tff(fact_6088_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% choose_dvd
tff(fact_6089_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% choose_dvd
tff(fact_6090_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% choose_dvd
tff(fact_6091_zdvd__mult__cancel1,axiom,
    ! [M: int,Na: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)),M)
      <=> ( aa_int_int(abs_abs_int,Na) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6092_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6093_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6094_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6095_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6096_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6097_signed__take__bit__eq__concat__bit,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(bit_concat_bit(Na,K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_6098_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6099_exp__eq__0__imp__not__bit,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6100_bit__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,Na))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Na) ) ).

% bit_Suc
tff(fact_6101_bit__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na) ) ).

% bit_Suc
tff(fact_6102_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,Na: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6103_stable__imp__bit__iff__odd,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6104_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,Na: 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_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6105_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) )
     => ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6106_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(A),N)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6107_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6108_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = aa_nat_int(bit_se2000444600071755411sk_int,Na) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Na),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_6109_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M2)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N) ) ) ) ).

% int_bit_bound
tff(fact_6110_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),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,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_6111_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(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_6112_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(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)))
    <=> aa_int_o(aa_int_fun_int_o(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_6113_bit__iff__odd,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na))) ) ).

% bit_iff_odd
tff(fact_6114_bit__iff__odd,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% bit_iff_odd
tff(fact_6115_bit__iff__odd,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(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))),Na))) ) ).

% bit_iff_odd
tff(fact_6116_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,Na))),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))),Na))) ).

% square_fact_le_2_fact
tff(fact_6117_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% Suc_mask_eq_exp
tff(fact_6118_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% mask_nat_less_exp
tff(fact_6119_bit__int__def,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% bit_int_def
tff(fact_6120_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na) )
         => aa_int_o(aa_int_fun_int_o(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,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6121_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6122_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6123_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6124_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6125_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_real(semiri2265585572941072030t_real,M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6126_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6127_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(semiri5044797733671781792omplex,Na) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6128_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6129_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6130_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(semiri2265585572941072030t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6131_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(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)),D))) ) ).

% incr_lemma
tff(fact_6132_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(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)),D))),Z) ) ).

% decr_lemma
tff(fact_6133_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),Na) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,Na)) ).

% pochhammer_same
tff(fact_6134_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),Na) = 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)),Na)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% pochhammer_same
tff(fact_6135_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(semiri5044797733671781792omplex,Na)) ).

% pochhammer_same
tff(fact_6136_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na))),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% pochhammer_same
tff(fact_6137_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% pochhammer_same
tff(fact_6138_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Na)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_6139_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Na),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_6140_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_6141_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Na)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_6142_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Na),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_6143_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_6144_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Na))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6145_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6146_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Na))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6147_even__bit__succ__iff,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A)),Na)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6148_even__bit__succ__iff,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Na)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6149_even__bit__succ__iff,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6150_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,Na: nat] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6151_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Na: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6152_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),Na)
          | ( Na = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6153_mask__nat__def,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = 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))),Na)),one_one_nat) ).

% mask_nat_def
tff(fact_6154_mask__half__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),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,Na),one_one_nat)) ).

% mask_half_int
tff(fact_6155_mask__int__def,axiom,
    ! [Na: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Na) = 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))),Na)),one_one_int) ).

% mask_int_def
tff(fact_6156_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(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,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6157_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6158_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_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))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6159_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),Na)
      <=> $ite(Na = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)),Na)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6160_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(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_6161_bit__rec,axiom,
    ! [A: code_integer,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_6162_bit__rec,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_6163_bit__rec,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
    <=> $ite(Na = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_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_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% bit_rec
tff(fact_6164_mask__eq__exp__minus__1,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = 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))),Na)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_6165_mask__eq__exp__minus__1,axiom,
    ! [Na: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Na) = 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))),Na)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_6166_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(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,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6167_set__bit__eq,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),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_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ).

% set_bit_eq
tff(fact_6168_unset__bit__eq,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),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_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ).

% unset_bit_eq
tff(fact_6169_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),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))),Na)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Na)))),aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% take_bit_Suc_from_most
tff(fact_6170_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),K) = aa_nat_int(bit_se2000444600071755411sk_int,Na) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_6171_sin__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(sin_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3))) ).

% sin_coeff_def
tff(fact_6172_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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,Na),K)),one_one_nat)),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_6173_cos__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(cos_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3)),zero_zero_real) ).

% cos_coeff_def
tff(fact_6174_fact__code,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_6175_fact__code,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,Na) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_6176_fact__code,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_6177_fact__code,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_6178_fact__code,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_6179_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(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,numeral_numeral_real(aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_6180_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),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(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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_6181_and_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),A) = A ).

% and.idem
tff(fact_6182_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_6183_and_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.left_idem
tff(fact_6184_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.left_idem
tff(fact_6185_and_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),B) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.right_idem
tff(fact_6186_and_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.right_idem
tff(fact_6187_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6188_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_6189_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_6190_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_6191_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6192_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6193_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% exp_less_cancel_iff
tff(fact_6194_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_6195_take__bit__and,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se2923211474154528505it_int(Na),A)),aa_int_int(bit_se2923211474154528505it_int(Na),B)) ).

% take_bit_and
tff(fact_6196_take__bit__and,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B)) ).

% take_bit_and
tff(fact_6197_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_6198_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_6199_bit_Oconj__one__right,axiom,
    ! [X: code_integer] : bit_se3949692690581998587nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ).

% bit.conj_one_right
tff(fact_6200_bit_Oconj__one__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).

% bit.conj_one_right
tff(fact_6201_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_6202_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_6203_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_6204_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_6205_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_6206_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_6207_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_6208_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_6209_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_6210_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ).

% and_numerals(2)
tff(fact_6211_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_6212_and__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_6213_and__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_6214_exp__less__one__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_6215_one__less__exp__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% one_less_exp_iff
tff(fact_6216_exp__le__one__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_6217_one__le__exp__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% one_le_exp_iff
tff(fact_6218_exp__ln,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_6219_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% exp_ln_iff
tff(fact_6220_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6221_and__numerals_I5_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))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6222_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_6223_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_6224_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_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_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_6225_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,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_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(3)
tff(fact_6226_and__minus__numerals_I2_J,axiom,
    ! [Na: 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,Na)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_6227_and__minus__numerals_I6_J,axiom,
    ! [Na: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_6228_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,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_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_6229_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,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_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(6)
tff(fact_6230_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_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(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,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_6231_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(4)
tff(fact_6232_and__minus__numerals_I5_J,axiom,
    ! [Na: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_6233_and__minus__numerals_I1_J,axiom,
    ! [Na: 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,Na)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_6234_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(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,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_6235_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_6236_of__nat__and__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_and_eq
tff(fact_6237_of__nat__and__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_and_eq
tff(fact_6238_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),C) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.assoc
tff(fact_6239_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.assoc
tff(fact_6240_and_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(bit_se725231765392027082nd_int(B),A) ).

% and.commute
tff(fact_6241_and_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),A) ).

% and.commute
tff(fact_6242_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(B),aa_int_int(bit_se725231765392027082nd_int(A),C)) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.left_commute
tff(fact_6243_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.left_commute
tff(fact_6244_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% exp_less_cancel
tff(fact_6245_bit__and__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),Na)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
        & aa_nat_o(bit_se1146084159140164899it_int(B),Na) ) ) ).

% bit_and_iff
tff(fact_6246_bit__and__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Na) ) ) ).

% bit_and_iff
tff(fact_6247_exp__not__eq__zero,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_6248_exp__not__eq__zero,axiom,
    ! [X: real] : aa_real_real(exp_real,X) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_6249_bit__and__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(K),L)),Na)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Na) ) ) ).

% bit_and_int_iff
tff(fact_6250_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_6251_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_6252_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_6253_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_6254_not__exp__less__zero,axiom,
    ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_6255_exp__gt__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,X)) ).

% exp_gt_zero
tff(fact_6256_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] : aa_real_real(exp_real,X2) = Y ) ).

% exp_total
tff(fact_6257_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_6258_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_6259_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Y) ) ).

% AND_upper2
tff(fact_6260_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),X) ) ).

% AND_upper1
tff(fact_6261_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(X),Y)) ) ).

% AND_lower
tff(fact_6262_real__sgn__eq,axiom,
    ! [X: real] : aa_real_real(sgn_sgn_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(abs_abs_real,X)) ).

% real_sgn_eq
tff(fact_6263_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_6264_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( 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(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_6265_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% mult_exp_exp
tff(fact_6266_mult__exp__exp,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% mult_exp_exp
tff(fact_6267_exp__diff,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_6268_exp__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_6269_take__bit__eq__mask,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Na),A) = aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% take_bit_eq_mask
tff(fact_6270_take__bit__eq__mask,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% take_bit_eq_mask
tff(fact_6271_sin__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% sin_coeff_Suc
tff(fact_6272_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Na)) ).

% not_bit_Suc_0_numeral
tff(fact_6273_exp__gt__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,X)) ) ).

% exp_gt_one
tff(fact_6274_exp__ge__add__one__self,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_6275_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_6276_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_6277_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K) ) ).

% and_less_eq
tff(fact_6278_exp__minus__inverse,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) = one_one_real ).

% exp_minus_inverse
tff(fact_6279_exp__minus__inverse,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) = one_one_complex ).

% exp_minus_inverse
tff(fact_6280_exp__of__nat__mult,axiom,
    ! [Na: nat,X: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)) = aa_nat_real(power_power_real(aa_real_real(exp_real,X)),Na) ).

% exp_of_nat_mult
tff(fact_6281_exp__of__nat__mult,axiom,
    ! [Na: nat,X: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),X)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),Na) ).

% exp_of_nat_mult
tff(fact_6282_exp__of__nat2__mult,axiom,
    ! [X: real,Na: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_nat_real(power_power_real(aa_real_real(exp_real,X)),Na) ).

% exp_of_nat2_mult
tff(fact_6283_exp__of__nat2__mult,axiom,
    ! [X: complex,Na: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(X),aa_nat_complex(semiri8010041392384452111omplex,Na))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),Na) ).

% exp_of_nat2_mult
tff(fact_6284_cos__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% cos_coeff_Suc
tff(fact_6285_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6286_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6287_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6288_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_6289_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(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_6290_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_6291_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_6292_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),X) ) ) ).

% ln_ge_iff
tff(fact_6293_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(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_6294_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6295_and__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6296_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6297_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6298_one__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6299_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6300_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one))) ).

% exp_le
tff(fact_6301_sgn__power__injE,axiom,
    ! [A: real,Na: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),Na)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B)),Na)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_6302_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),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_6303_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 )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,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_6304_exp__divide__power__eq,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_6305_exp__divide__power__eq,axiom,
    ! [Na: nat,X: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,Na)))),Na) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_6306_tanh__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,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% tanh_altdef
tff(fact_6307_tanh__altdef,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% tanh_altdef
tff(fact_6308_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6309_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6310_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_6311_bit__nat__def,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Na)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(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))),Na))) ) ).

% bit_nat_def
tff(fact_6312_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_6313_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)) = aa_nat_real(power_power_real(aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_6314_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_6315_exp__bound__half,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_6316_exp__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_6317_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_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_6318_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))) ) ) ).

% real_exp_bound_lemma
tff(fact_6319_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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,Na)))),Na)),aa_real_real(exp_real,X)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_6320_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_6321_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_6322_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_6323_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,X)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_6324_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,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,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_6325_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(modulo_modulo_int(K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,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(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_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(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_6326_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_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)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_6327_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,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,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_6328_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,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),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),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_6329_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_6330_nat__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Na)) = Na ).

% nat_int
tff(fact_6331_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_6332_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_6333_nat__of__bool,axiom,
    ! [P: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% nat_of_bool
tff(fact_6334_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_6335_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_6336_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_6337_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_6338_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_6339_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_6340_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_6341_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_6342_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_6343_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_6344_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6345_nat__zminus__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_6346_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_6347_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_6348_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_6349_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_6350_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_6351_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_6352_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_6353_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_6354_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_6355_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_6356_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_6357_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_6358_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ).

% dvd_nat_abs_iff
tff(fact_6359_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_abs_dvd_iff
tff(fact_6360_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_6361_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(power_power_real(A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_6362_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_6363_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_6364_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_6365_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_6366_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_6367_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_6368_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_6369_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_6370_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_6371_and__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% and_nat_def
tff(fact_6372_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_6373_nat__numeral__as__int,axiom,
    ! [X3: num] : aa_num_nat(numeral_numeral_nat,X3) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X3)) ).

% nat_numeral_as_int
tff(fact_6374_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_6375_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_6376_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_6377_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(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_6378_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_6379_log__def,axiom,
    ! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_6380_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_6381_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_6382_unset__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),Na) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% unset_bit_nat_def
tff(fact_6383_nat__mask__eq,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Na)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Na) ).

% nat_mask_eq
tff(fact_6384_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_6385_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_6386_nat__le__iff,axiom,
    ! [X: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_6387_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_6388_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_6389_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_6390_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_6391_log__ln,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),X) ).

% log_ln
tff(fact_6392_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6393_less__log__of__power,axiom,
    ! [B: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(B),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M)) ) ) ).

% less_log_of_power
tff(fact_6394_log__of__power__eq,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(power_power_real(B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_6395_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_6396_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_6397_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_6398_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_6399_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N4: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => aa_nat_o(P,N4) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_6400_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_6401_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(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_6402_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(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_6403_Suc__as__int,axiom,
    ! [X3: nat] : aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ).

% Suc_as_int
tff(fact_6404_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
     => ( aa_int_o(aa_int_fun_int_o(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_6405_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(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_6406_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(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_6407_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_6408_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_6409_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_6410_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(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_6411_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_6412_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_6413_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_6414_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_6415_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_6416_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_6417_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_6418_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Na)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ) ).

% bit_nat_iff
tff(fact_6419_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_6420_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(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_6421_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(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_6422_le__log__of__power,axiom,
    ! [B: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(B),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M)) ) ) ).

% le_log_of_power
tff(fact_6423_log__base__pow,axiom,
    ! [A: real,Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(power_power_real(A),Na)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_6424_log__nat__power,axiom,
    ! [X: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(power_power_real(X),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_nat_power
tff(fact_6425_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(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_6426_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_6427_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(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_6428_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      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)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_6429_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_6430_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_6431_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_6432_log2__of__power__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_6433_log__of__power__less,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_6434_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(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_6435_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_6436_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_6437_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_6438_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_6439_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_6440_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_6441_log__of__power__le,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_6442_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_6443_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_6444_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_6445_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = $ite(
        ( ( M = zero_zero_nat )
        | ( Na = zero_zero_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_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,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_6446_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_6447_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_6448_and__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ))),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_6449_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_6450_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_6451_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_6452_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_6453_sin__cos__npi,axiom,
    ! [Na: 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))),Na)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% sin_cos_npi
tff(fact_6454_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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,Na)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(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,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_6455_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))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6456_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(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,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(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,Na)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_6457_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,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,Na),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_6458_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(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,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(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,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_6459_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_6460_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_6461_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_6462_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_6463_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_6464_floor__zero,axiom,
    archim3151403230148437115or_rat(zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_6465_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_6466_floor__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_6467_ceiling__zero,axiom,
    archim2889992004027027881ng_rat(zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_6468_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_6469_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_6470_floor__one,axiom,
    archim3151403230148437115or_rat(one_one_rat) = one_one_int ).

% floor_one
tff(fact_6471_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_6472_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_6473_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_6474_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_6475_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_6476_cos__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi2
tff(fact_6477_cos__periodic__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi
tff(fact_6478_sin__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi2
tff(fact_6479_sin__periodic__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi
tff(fact_6480_zero__le__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_floor
tff(fact_6481_zero__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X) ) ).

% zero_le_floor
tff(fact_6482_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_6483_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_6484_floor__less__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% floor_less_zero
tff(fact_6485_floor__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_6486_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(V)),X) ) ).

% numeral_le_floor
tff(fact_6487_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(V)),X) ) ).

% numeral_le_floor
tff(fact_6488_zero__less__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ).

% zero_less_floor
tff(fact_6489_zero__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% zero_less_floor
tff(fact_6490_floor__le__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% floor_le_zero
tff(fact_6491_floor__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_le_zero
tff(fact_6492_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_6493_ceiling__le__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_6494_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(V)) ) ).

% floor_less_numeral
tff(fact_6495_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),numeral_numeral_rat(V)) ) ).

% floor_less_numeral
tff(fact_6496_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% zero_less_ceiling
tff(fact_6497_zero__less__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% zero_less_ceiling
tff(fact_6498_one__le__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ).

% one_le_floor
tff(fact_6499_one__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% one_le_floor
tff(fact_6500_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),numeral_numeral_rat(V)) ) ).

% ceiling_le_numeral
tff(fact_6501_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_6502_floor__less__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% floor_less_one
tff(fact_6503_floor__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_less_one
tff(fact_6504_ceiling__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_6505_ceiling__less__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_6506_one__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% one_le_ceiling
tff(fact_6507_one__le__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% one_le_ceiling
tff(fact_6508_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(V)),X) ) ).

% numeral_less_ceiling
tff(fact_6509_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(V)),X) ) ).

% numeral_less_ceiling
tff(fact_6510_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_6511_floor__neg__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_6512_ceiling__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ).

% ceiling_le_one
tff(fact_6513_ceiling__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% ceiling_le_one
tff(fact_6514_one__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ).

% one_less_ceiling
tff(fact_6515_one__less__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ).

% one_less_ceiling
tff(fact_6516_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(V))) = aa_int_int(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_6517_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_6518_floor__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),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_6519_floor__diff__numeral,axiom,
    ! [X: rat,V: num] : archim3151403230148437115or_rat(minus_minus_rat(X,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_6520_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_6521_ceiling__neg__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_6522_ceiling__add__one,axiom,
    ! [X: rat] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),one_one_int) ).

% ceiling_add_one
tff(fact_6523_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_6524_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_6525_floor__diff__one,axiom,
    ! [X: rat] : archim3151403230148437115or_rat(minus_minus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(X)),one_one_int) ).

% floor_diff_one
tff(fact_6526_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),numeral_numeral_real(V))) = aa_int_int(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_6527_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(X,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_6528_ceiling__diff__one,axiom,
    ! [X: rat] : archim2889992004027027881ng_rat(minus_minus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),one_one_int) ).

% ceiling_diff_one
tff(fact_6529_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_6530_floor__numeral__power,axiom,
    ! [X: num,Na: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(power_power_real(numeral_numeral_real(X)),Na)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ).

% floor_numeral_power
tff(fact_6531_floor__numeral__power,axiom,
    ! [X: num,Na: nat] : archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ).

% floor_numeral_power
tff(fact_6532_ceiling__numeral__power,axiom,
    ! [X: num,Na: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(power_power_real(numeral_numeral_real(X)),Na)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ).

% ceiling_numeral_power
tff(fact_6533_ceiling__numeral__power,axiom,
    ! [X: num,Na: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ).

% ceiling_numeral_power
tff(fact_6534_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_6535_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_6536_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),uminus_uminus_rat(one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_6537_ceiling__less__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_6538_zero__le__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X) ) ).

% zero_le_ceiling
tff(fact_6539_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),X) ) ).

% zero_le_ceiling
tff(fact_6540_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_6541_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)),X) ) ).

% numeral_less_floor
tff(fact_6542_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)),X) ) ).

% numeral_less_floor
tff(fact_6543_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_6544_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_6545_one__less__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_6546_one__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_6547_floor__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_6548_floor__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),numeral_numeral_rat(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_6549_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_6550_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_6551_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)),X) ) ).

% numeral_le_ceiling
tff(fact_6552_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),X) ) ).

% numeral_le_ceiling
tff(fact_6553_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X) ) ).

% neg_numeral_le_floor
tff(fact_6554_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(V))),X) ) ).

% neg_numeral_le_floor
tff(fact_6555_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% floor_less_neg_numeral
tff(fact_6556_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% floor_less_neg_numeral
tff(fact_6557_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6558_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6559_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_6560_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_6561_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_6562_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_6563_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_6564_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_6565_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_6566_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_6567_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_6568_cos__npi2,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% cos_npi2
tff(fact_6569_cos__npi,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% cos_npi
tff(fact_6570_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_6571_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_6572_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_6573_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_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(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_6574_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_6575_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_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(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_6576_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_6577_sin__2npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_6578_cos__2npi,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)) = one_one_real ).

% cos_2npi
tff(fact_6579_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_6580_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X) ) ).

% neg_numeral_less_floor
tff(fact_6581_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(V))),one_one_rat)),X) ) ).

% neg_numeral_less_floor
tff(fact_6582_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_6583_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_6584_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_6585_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_6586_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_6587_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_6588_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_6589_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_6590_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_6591_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_6592_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_6593_floor__le__ceiling,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim7802044766580827645g_real,X)) ).

% floor_le_ceiling
tff(fact_6594_floor__le__ceiling,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),archim2889992004027027881ng_rat(X)) ).

% floor_le_ceiling
tff(fact_6595_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_6596_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_6597_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_6598_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_6599_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_6600_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(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_6601_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),archim3151403230148437115or_rat(X))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_6602_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_6603_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_6604_sin__double,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,X))),aa_real_real(cos_real,X)) ).

% sin_double
tff(fact_6605_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_6606_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_6607_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y)) ) ).

% floor_mono
tff(fact_6608_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% floor_less_cancel
tff(fact_6609_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% floor_less_cancel
tff(fact_6610_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Y)),archim2889992004027027881ng_rat(X)) ) ).

% ceiling_mono
tff(fact_6611_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X)) ) ).

% ceiling_mono
tff(fact_6612_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),archim2889992004027027881ng_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% ceiling_less_cancel
tff(fact_6613_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% ceiling_less_cancel
tff(fact_6614_sin__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real) ).

% sin_le_one
tff(fact_6615_cos__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real) ).

% cos_le_one
tff(fact_6616_sin__cos__le1,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(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_6617_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_6618_cos__squared__eq,axiom,
    ! [X: real] : aa_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(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_6619_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_6620_sin__squared__eq,axiom,
    ! [X: real] : aa_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(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_6621_le__floor__add,axiom,
    ! [X: real,Y: real] : aa_int_o(aa_int_fun_int_o(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_6622_le__floor__add,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y))),archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))) ).

% le_floor_add
tff(fact_6623_of__nat__ceiling,axiom,
    ! [R4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R4),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,archim2889992004027027881ng_rat(R4)))) ).

% of_nat_ceiling
tff(fact_6624_of__nat__ceiling,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R4),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R4)))) ).

% of_nat_ceiling
tff(fact_6625_sin__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero
tff(fact_6626_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),archim2889992004027027881ng_rat(Y))) ).

% ceiling_add_le
tff(fact_6627_ceiling__add__le,axiom,
    ! [X: real,Y: real] : aa_int_o(aa_int_fun_int_o(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_6628_real__nat__ceiling__ge,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_6629_sin__ge__minus__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_6630_cos__ge__minus__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_6631_abs__sin__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_6632_abs__cos__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_6633_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_6634_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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)),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)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_6635_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_6636_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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)),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)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_6637_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_6638_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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)),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)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_6639_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_6640_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_6641_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_6642_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_6643_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_6644_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_6645_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_6646_cos__double,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_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(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_6647_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_6648_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_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_6649_of__nat__floor,axiom,
    ! [R4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R4)))),R4) ) ).

% of_nat_floor
tff(fact_6650_of__nat__floor,axiom,
    ! [R4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,archim3151403230148437115or_rat(R4)))),R4) ) ).

% of_nat_floor
tff(fact_6651_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_6652_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),one_one_int) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) ).

% one_add_floor
tff(fact_6653_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : aa_nat_o(aa_nat_fun_nat_o(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_6654_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,archim3151403230148437115or_rat(A))),aa_int_nat(nat2,archim3151403230148437115or_rat(B)))),aa_int_nat(nat2,archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B)))) ).

% le_mult_nat_floor
tff(fact_6655_floor__divide__of__nat__eq,axiom,
    ! [M: nat,Na: 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,Na))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) ).

% floor_divide_of_nat_eq
tff(fact_6656_floor__divide__of__nat__eq,axiom,
    ! [M: nat,Na: nat] : archim3151403230148437115or_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,Na))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) ).

% floor_divide_of_nat_eq
tff(fact_6657_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6658_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_6659_floor__eq3,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = Na ) ) ) ).

% floor_eq3
tff(fact_6660_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_6661_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_6662_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_6663_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),pi)
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_6664_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6665_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),pi)
            & ( X = aa_real_real(cos_real,T) )
            & ( Y = aa_real_real(sin_real,T) ) ) ) ) ).

% sincos_total_pi
tff(fact_6666_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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(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_6667_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(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)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_6668_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(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_6669_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B))) ) ) ).

% le_mult_floor
tff(fact_6670_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(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)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_6671_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero_02
tff(fact_6672_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B))) ) ) ).

% mult_ceiling_le
tff(fact_6673_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(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_6674_floor__eq4,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = Na ) ) ) ).

% floor_eq4
tff(fact_6675_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_6676_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_6677_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),numeral_numeral_real(aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),numeral_numeral_real(aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X2 ) ) ) ).

% cos_is_zero
tff(fact_6678_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_6679_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),pi)
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
                  & ( aa_real_real(cos_real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_6680_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
              & ( X = aa_real_real(cos_real,T) )
              & ( Y = aa_real_real(sin_real,T) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_6681_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
          & ( X = aa_real_real(cos_real,T) )
          & ( Y = aa_real_real(sin_real,T) ) ) ) ).

% sincos_total_2pi_le
tff(fact_6682_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( ( X = aa_real_real(cos_real,T) )
               => ( Y != aa_real_real(sin_real,T) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_6683_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(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,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_6684_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_45
tff(fact_6685_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_45
tff(fact_6686_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_6687_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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)),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)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_6688_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_6689_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_6690_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero2
tff(fact_6691_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_6692_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_6693_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_30
tff(fact_6694_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero
tff(fact_6695_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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_6696_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_6697_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_6698_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_60
tff(fact_6699_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_60
tff(fact_6700_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_30
tff(fact_6701_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_6702_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_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_6703_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_6704_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_6705_sin__le__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_6706_sin__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_6707_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_6708_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_6709_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_6710_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_6711_cos__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_ge_zero
tff(fact_6712_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X4: 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,X4)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
        | ? [X4: 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,X4)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_6713_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(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,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_6714_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_6715_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_6716_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_6717_floor__log__nat__eq__if,axiom,
    ! [B: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(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,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(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,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_6718_sin__zero__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_6719_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_6720_cos__zero__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_6721_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_6722_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_6723_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_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,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(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_6724_sin__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_6725_cos__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_6726_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_6727_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(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) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),X)
            & aa_real_o(aa_real_fun_real_o(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_6728_cos__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_6729_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_6730_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_6731_powr__one__eq__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),A) = one_one_real ).

% powr_one_eq_one
tff(fact_6732_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_6733_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_6734_powr__zero__eq__one,axiom,
    ! [X: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = $ite(X = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_6735_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6736_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_6737_tan__periodic__pi,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(tan_real,X) ).

% tan_periodic_pi
tff(fact_6738_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6739_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% powr_one_gt_zero_iff
tff(fact_6740_powr__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6741_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_6742_numeral__powr__numeral__real,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(M)),numeral_numeral_real(Na)) = aa_nat_real(power_power_real(numeral_numeral_real(M)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_powr_numeral_real
tff(fact_6743_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(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_6744_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6745_tan__periodic__n,axiom,
    ! [X: real,Na: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(Na)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_6746_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_6747_tan__periodic__nat,axiom,
    ! [X: real,Na: 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,Na)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_6748_norm__cos__sin,axiom,
    ! [Tb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,Tb)),aa_real_real(sin_real,Tb))) = one_one_real ).

% norm_cos_sin
tff(fact_6749_powr__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),numeral_numeral_real(Na)) = aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,Na)) ) ) ).

% powr_numeral
tff(fact_6750_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_6751_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_6752_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_6753_square__powr__half,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_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),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_6754_powr__non__neg,axiom,
    ! [A: real,X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real) ).

% powr_non_neg
tff(fact_6755_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(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_6756_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_6757_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(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_6758_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(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_6759_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_6760_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_6761_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).

% complex_add
tff(fact_6762_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(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_6763_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(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_6764_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) ) ) ).

% gr_one_powr
tff(fact_6765_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6766_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_6767_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
           => aa_real_o(aa_real_fun_real_o(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_6768_powr__le1,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_6769_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(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_6770_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_6771_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_6772_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_6773_powr__add,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)) ).

% powr_add
tff(fact_6774_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z22)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z22)) ).

% powr_diff
tff(fact_6775_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_6776_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_6777_arcsin__minus,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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_6778_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(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_6779_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_6780_tan__def,axiom,
    ! [X3: complex] : aa_complex_complex(tan_complex,X3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X3)),aa_complex_complex(cos_complex,X3)) ).

% tan_def
tff(fact_6781_tan__def,axiom,
    ! [X3: real] : aa_real_real(tan_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X3)),aa_real_real(cos_real,X3)) ).

% tan_def
tff(fact_6782_powr__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(power_power_real(X),Na) ) ) ).

% powr_realpow
tff(fact_6783_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% powr_less_iff
tff(fact_6784_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) ) ) ) ).

% less_powr_iff
tff(fact_6785_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_less_iff
tff(fact_6786_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X) ) ) ) ).

% less_log_iff
tff(fact_6787_powr__minus__divide,axiom,
    ! [X: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)) ).

% powr_minus_divide
tff(fact_6788_powr__neg__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6789_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_6790_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_6791_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_6792_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% powr_le_iff
tff(fact_6793_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) ) ) ) ).

% le_powr_iff
tff(fact_6794_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_6795_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X) ) ) ) ).

% le_log_iff
tff(fact_6796_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(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_6797_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(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_6798_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_6799_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(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_6800_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(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_6801_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(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_6802_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_6803_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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_6804_powr__def,axiom,
    ! [X: real,A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A) = $ite(X = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,X)))) ).

% powr_def
tff(fact_6805_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X2)) ) ) ).

% lemma_tan_total
tff(fact_6806_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_gt_zero
tff(fact_6807_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y3) = Y ) )
         => ( Y3 = X2 ) ) ) ).

% tan_total
tff(fact_6808_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X)) ) ) ) ).

% tan_monotone
tff(fact_6809_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_6810_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_6811_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_6812_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_6813_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_6814_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(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_6815_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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_6816_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_6817_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_6818_powr__half__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_6819_powr__neg__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,Na))) ) ) ).

% powr_neg_numeral
tff(fact_6820_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_pos_pi2_le
tff(fact_6821_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_6822_tan__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_6823_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_6824_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_6825_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_6826_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_6827_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_6828_arctan__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_6829_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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_6830_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_6831_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_6832_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_6833_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_6834_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_6835_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_6836_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_6837_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_6838_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_6839_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_6840_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_6841_arcsin__sin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_6842_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_6843_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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,numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_6844_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,X))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),X) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_6845_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_6846_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(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_6847_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_6848_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(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),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_6849_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_6850_sin__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_6851_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(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),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_6852_cos__npi__int,axiom,
    ! [Na: int] :
      aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Na))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_6853_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_6854_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_6855_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) = X )
    <=> ? [N4: int] : X = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_floor_cancel
tff(fact_6856_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X)) = X )
    <=> ? [N4: int] : X = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_floor_cancel
tff(fact_6857_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)) = X )
    <=> ? [N4: int] : X = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_ceiling_cancel
tff(fact_6858_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)) = X )
    <=> ? [N4: int] : X = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_ceiling_cancel
tff(fact_6859_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_6860_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_6861_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_6862_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_6863_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_6864_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_6865_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_6866_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_6867_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_6868_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_6869_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_6870_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_6871_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_6872_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_6873_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_6874_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_6875_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_6876_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_6877_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_6878_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_6879_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_6880_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_6881_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_6882_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6883_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6884_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6885_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Na) ) ) ).

% of_int_eq_numeral_iff
tff(fact_6886_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_6887_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_6888_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_6889_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_6890_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_6891_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_6892_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_6893_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_6894_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_6895_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_6896_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_6897_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_6898_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_6899_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_6900_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_6901_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_6902_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(aa_rat_fun_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_6903_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_6904_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = one_one_complex )
    <=> ( X = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_6905_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_6906_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_6907_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_6908_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_6909_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_6910_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_6911_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_6912_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% of_real_numeral
tff(fact_6913_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,numeral_numeral_real(W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_6914_of__real__divide,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_6915_of__real__divide,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_divide
tff(fact_6916_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_6917_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)) = 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_6918_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_6919_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% of_int_of_nat_eq
tff(fact_6920_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% of_int_of_nat_eq
tff(fact_6921_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_complex(semiri8010041392384452111omplex,Na) ).

% of_int_of_nat_eq
tff(fact_6922_of__int__of__nat__eq,axiom,
    ! [Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% of_int_of_nat_eq
tff(fact_6923_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_6924_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_6925_of__real__power,axiom,
    ! [X: real,Na: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(power_power_real(X),Na)) = aa_nat_real(power_power_real(aa_real_real(real_V1803761363581548252l_real,X)),Na) ).

% of_real_power
tff(fact_6926_of__real__power,axiom,
    ! [X: real,Na: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(power_power_real(X),Na)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,X)),Na) ).

% of_real_power
tff(fact_6927_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_6928_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_6929_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_6930_of__int__abs,axiom,
    ! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = abs_abs_rat(aa_int_rat(ring_1_of_int_rat,X)) ).

% of_int_abs
tff(fact_6931_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Na)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Na) ).

% of_int_power
tff(fact_6932_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Na)) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,Z)),Na) ).

% of_int_power
tff(fact_6933_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Na)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Na) ).

% of_int_power
tff(fact_6934_of__int__power,axiom,
    ! [Z: int,Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Na)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Na) ).

% of_int_power
tff(fact_6935_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_6936_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_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_6937_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_6938_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_6939_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_6940_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(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_6941_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_6942_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_6943_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_6944_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_real(ring_1_of_int_real,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_int_of_bool
tff(fact_6945_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_rat(ring_1_of_int_rat,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n2052037380579107095ol_rat((P)) ).

% of_int_of_bool
tff(fact_6946_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_int(ring_1_of_int_int,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_int_of_bool
tff(fact_6947_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_Code_integer(zero_n356916108424825756nteger,(P)) ).

% of_int_of_bool
tff(fact_6948_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_6949_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_6950_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_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,archim2889992004027027881ng_rat(X)),Z) ).

% ceiling_add_of_int
tff(fact_6951_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_6952_of__nat__nat__take__bit__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K))) = aa_int_real(ring_1_of_int_real,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6953_of__nat__nat__take__bit__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K))) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6954_of__nat__nat__take__bit__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K))) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6955_of__nat__nat__take__bit__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Na),K))) = aa_int_rat(ring_1_of_int_rat,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6956_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_6957_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6958_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6959_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6960_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6961_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6962_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6963_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6964_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6965_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6966_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6967_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6968_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6969_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6970_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6971_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6972_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6973_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6974_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6975_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6976_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6977_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6978_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6979_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6980_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6981_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6982_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6983_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6984_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6985_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6986_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6987_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6988_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6989_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6990_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6991_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6992_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6993_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) ).

% of_real_neg_numeral
tff(fact_6994_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_6995_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6996_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6997_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6998_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6999_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_7000_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_7001_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),Na) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7002_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(numeral_numeral_real(X)),Na) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7003_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7004_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7005_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7006_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(numeral_numeral_real(X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7007_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7008_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7009_cos__of__real__pi,axiom,
    aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_of_real_pi
tff(fact_7010_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_7011_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_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_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_7012_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_7013_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_7014_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_7015_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_7016_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_7017_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(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_7018_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(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_7019_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(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_7020_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(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_7021_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_7022_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_7023_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7024_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7025_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(numeral_numeral_real(X)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7026_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7027_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7028_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(numeral_numeral_real(X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7029_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(numeral_numeral_real(X)),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7030_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7031_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7032_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(numeral_numeral_real(X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7033_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7034_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7035_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),Na) = 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))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7036_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) = 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))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7037_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),Na) = 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))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7038_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na) = 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))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7039_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),Na) = 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))),Na) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7040_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7041_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: 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))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7042_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7043_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7044_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7045_norm__of__real__add1,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_7046_norm__of__real__add1,axiom,
    ! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_7047_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),numeral_numeral_real(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_7048_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_7049_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_7050_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_7051_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_7052_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_7053_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_7054_sin__int__2pin,axiom,
    ! [Na: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Na))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7055_cos__int__2pin,axiom,
    ! [Na: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Na))) = one_one_real ).

% cos_int_2pin
tff(fact_7056_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7057_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7058_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7059_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7060_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7061_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7062_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7063_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7064_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7065_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(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))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7066_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7067_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7068_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),Na)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7069_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7070_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7071_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Na: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),Na)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Na)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7072_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_7073_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_7074_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X) ).

% ex_of_int_less
tff(fact_7075_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X) ).

% ex_of_int_less
tff(fact_7076_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_7077_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_7078_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_7079_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_7080_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_7081_sgn__eq,axiom,
    ! [Z: complex] : aa_complex_complex(sgn_sgn_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% sgn_eq
tff(fact_7082_of__int__floor__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_7083_of__int__floor__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X))),X) ).

% of_int_floor_le
tff(fact_7084_le__of__int__ceiling,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X))) ).

% le_of_int_ceiling
tff(fact_7085_le__of__int__ceiling,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_7086_take__bit__of__int,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)) ).

% take_bit_of_int
tff(fact_7087_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se725231765392027082nd_int(K),L)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_and_eq
tff(fact_7088_of__int__mask__eq,axiom,
    ! [Na: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)) = aa_nat_int(bit_se2000444600071755411sk_int,Na) ).

% of_int_mask_eq
tff(fact_7089_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_7090_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_7091_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X) ) ).

% le_floor_iff
tff(fact_7092_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ) ).

% le_floor_iff
tff(fact_7093_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_7094_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_7095_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),A) ) ).

% ceiling_le
tff(fact_7096_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A) ) ).

% ceiling_le
tff(fact_7097_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_7098_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_7099_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ) ).

% less_ceiling_iff
tff(fact_7100_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X) ) ).

% less_ceiling_iff
tff(fact_7101_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_7102_floor__add__int,axiom,
    ! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),Z) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_7103_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_7104_int__add__floor,axiom,
    ! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),archim3151403230148437115or_rat(X)) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ).

% int_add_floor
tff(fact_7105_real__of__int__div4,axiom,
    ! [Na: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,X))) ).

% real_of_int_div4
tff(fact_7106_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(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_7107_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : archim3151403230148437115or_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(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_7108_floor__power,axiom,
    ! [X: real,Na: nat] :
      ( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(power_power_real(X),Na)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,X)),Na) ) ) ).

% floor_power
tff(fact_7109_floor__power,axiom,
    ! [X: rat,Na: nat] :
      ( ( X = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X)) )
     => ( archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(X),Na)) = aa_nat_int(power_power_int(archim3151403230148437115or_rat(X)),Na) ) ) ).

% floor_power
tff(fact_7110_real__of__int__div,axiom,
    ! [D: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Na)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_7111_complex__of__real__add__Complex,axiom,
    ! [R4: real,X: real,Y: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R4)),aa_real_complex(complex2(X),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R4),X)),Y) ).

% complex_of_real_add_Complex
tff(fact_7112_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R4: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(X),Y)),aa_real_complex(real_V4546457046886955230omplex,R4)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),R4)),Y) ).

% Complex_add_complex_of_real
tff(fact_7113_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_le_arccos
tff(fact_7114_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ) ) ).

% arccos_le_mono
tff(fact_7115_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(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_7116_norm__less__p1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(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_7117_norm__less__p1,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(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_7118_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_7119_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_7120_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_7121_of__int__leD,axiom,
    ! [Na: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).

% of_int_leD
tff(fact_7122_of__int__leD,axiom,
    ! [Na: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% of_int_leD
tff(fact_7123_of__int__leD,axiom,
    ! [Na: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).

% of_int_leD
tff(fact_7124_of__int__leD,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% of_int_leD
tff(fact_7125_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_7126_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_7127_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_7128_of__int__lessD,axiom,
    ! [Na: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).

% of_int_lessD
tff(fact_7129_of__int__lessD,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% of_int_lessD
tff(fact_7130_of__int__lessD,axiom,
    ! [Na: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% of_int_lessD
tff(fact_7131_of__int__lessD,axiom,
    ! [Na: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Na))),X)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).

% of_int_lessD
tff(fact_7132_floor__exists1,axiom,
    ! [X: rat] :
    ? [X2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),X)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int))) )
         => ( Y3 = X2 ) ) ) ).

% floor_exists1
tff(fact_7133_floor__exists1,axiom,
    ! [X: real] :
    ? [X2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int))) )
         => ( Y3 = X2 ) ) ) ).

% floor_exists1
tff(fact_7134_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)
      & aa_rat_o(aa_rat_fun_rat_o(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_7135_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X)
      & aa_real_o(aa_real_fun_real_o(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_7136_of__int__ceiling__le__add__one,axiom,
    ! [R4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R4))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R4),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_7137_of__int__ceiling__le__add__one,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R4))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R4),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_7138_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_7139_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_7140_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_7141_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_7142_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus_uminus_rat(numeral_numeral_rat(K)) ).

% of_int_neg_numeral
tff(fact_7143_of__int__ceiling__diff__one__le,axiom,
    ! [R4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R4)),one_one_rat)),R4) ).

% of_int_ceiling_diff_one_le
tff(fact_7144_of__int__ceiling__diff__one__le,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(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,R4))),one_one_real)),R4) ).

% of_int_ceiling_diff_one_le
tff(fact_7145_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_7146_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_7147_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_7148_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),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_7149_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_7150_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_7151_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : archim2889992004027027881ng_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_7152_ceiling__altdef,axiom,
    ! [X: real] :
      aa_real_int(archim7802044766580827645g_real,X) = $ite(X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_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_7153_ceiling__altdef,axiom,
    ! [X: rat] :
      archim2889992004027027881ng_rat(X) = $ite(X = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(X),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),one_one_int)) ).

% ceiling_altdef
tff(fact_7154_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: 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,K3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_7155_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(X),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_7156_floor__eq,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Na ) ) ) ).

% floor_eq
tff(fact_7157_real__of__int__floor__add__one__gt,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R4))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_7158_real__of__int__floor__add__one__ge,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R4))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_7159_real__of__int__floor__gt__diff__one,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R4),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R4))) ).

% real_of_int_floor_gt_diff_one
tff(fact_7160_real__of__int__floor__ge__diff__one,axiom,
    ! [R4: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R4),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R4))) ).

% real_of_int_floor_ge_diff_one
tff(fact_7161_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_7162_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_7163_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ).

% arccos_less_mono
tff(fact_7164_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_7165_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)
     => ( aa_real_o(aa_real_fun_real_o(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_7166_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( archim3151403230148437115or_rat(X) = Z ) ) ) ).

% floor_unique
tff(fact_7167_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X)
        & aa_real_o(aa_real_fun_real_o(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_7168_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( archim3151403230148437115or_rat(X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_7169_floor__split,axiom,
    ! [P: fun_int_o,Tb: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Tb))
    <=> ! [I3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I3)),Tb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Tb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I3)),one_one_real)) )
         => aa_int_o(P,I3) ) ) ).

% floor_split
tff(fact_7170_floor__split,axiom,
    ! [P: fun_int_o,Tb: rat] :
      ( aa_int_o(P,archim3151403230148437115or_rat(Tb))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I3)),Tb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Tb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)) )
         => aa_int_o(P,I3) ) ) ).

% floor_split
tff(fact_7171_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(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_7172_ceiling__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)),one_one_rat)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X))) ) ).

% ceiling_correct
tff(fact_7173_ceiling__correct,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_correct
tff(fact_7174_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))
       => ( archim2889992004027027881ng_rat(X) = Z ) ) ) ).

% ceiling_unique
tff(fact_7175_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(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)
     => ( aa_real_o(aa_real_fun_real_o(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_7176_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( archim2889992004027027881ng_rat(X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_7177_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(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)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_7178_ceiling__split,axiom,
    ! [P: fun_int_o,Tb: rat] :
      ( aa_int_o(P,archim2889992004027027881ng_rat(Tb))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I3),one_one_rat)),Tb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Tb),aa_int_rat(ring_1_of_int_rat,I3)) )
         => aa_int_o(P,I3) ) ) ).

% ceiling_split
tff(fact_7179_ceiling__split,axiom,
    ! [P: fun_int_o,Tb: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Tb))
    <=> ! [I3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I3)),one_one_real)),Tb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Tb),aa_int_real(ring_1_of_int_real,I3)) )
         => aa_int_o(P,I3) ) ) ).

% ceiling_split
tff(fact_7180_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(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_7181_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim3151403230148437115or_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X) ) ).

% less_floor_iff
tff(fact_7182_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(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_7183_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_7184_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_7185_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z)
    <=> aa_real_o(aa_real_fun_real_o(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_7186_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X) ) ).

% le_ceiling_iff
tff(fact_7187_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(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_7188_floor__correct,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X)
      & aa_real_o(aa_real_fun_real_o(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_7189_floor__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X))),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),one_one_int))) ) ).

% floor_correct
tff(fact_7190_real__of__int__div2,axiom,
    ! [Na: int,X: int] : aa_real_o(aa_real_fun_real_o(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,Na)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),X)))) ).

% real_of_int_div2
tff(fact_7191_real__of__int__div3,axiom,
    ! [Na: int,X: int] : aa_real_o(aa_real_fun_real_o(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,Na)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),X)))),one_one_real) ).

% real_of_int_div3
tff(fact_7192_floor__eq2,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Na ) ) ) ).

% floor_eq2
tff(fact_7193_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_7194_floor__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(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_7195_floor__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P3,Q3)))),Q3)),P3) ) ).

% floor_divide_lower
tff(fact_7196_ceiling__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P3,Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_7197_ceiling__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(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_7198_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_7199_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_7200_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_7201_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_7202_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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_7203_arccos__minus,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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_7204_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7205_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7206_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7207_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7208_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7209_floor__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(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_7210_floor__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P3,Q3)))),one_one_rat)),Q3)) ) ).

% floor_divide_upper
tff(fact_7211_ceiling__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(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_7212_ceiling__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P3,Q3))),one_one_rat)),Q3)),P3) ) ).

% ceiling_divide_lower
tff(fact_7213_ceiling__eq,axiom,
    ! [Na: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Na)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Na)),one_one_rat))
       => ( archim2889992004027027881ng_rat(X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_7214_ceiling__eq,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_7215_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X4: 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,X4)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7216_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_7217_arccos__minus__abs,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_7218_sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(sin_real,X) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)) ).

% sin_cos_eq
tff(fact_7219_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_7220_cos__sin__eq,axiom,
    ! [X: real] : aa_real_real(cos_real,X) = aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)) ).

% cos_sin_eq
tff(fact_7221_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_7222_minus__sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_7223_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_7224_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_7225_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),X)
            & aa_real_o(aa_real_fun_real_o(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_7226_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I3: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3)
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7227_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3)
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7228_powr__int,axiom,
    ! [X: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(power_power_real(X),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_7229_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
       => ( archim7778729529865785530nd_rat(X) = Y ) ) ) ).

% round_unique
tff(fact_7230_round__unique,axiom,
    ! [X: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_7231_round__unique_H,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(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,Na)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = Na ) ) ).

% round_unique'
tff(fact_7232_round__unique_H,axiom,
    ! [X: rat,Na: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,Na)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))
     => ( archim7778729529865785530nd_rat(X) = Na ) ) ).

% round_unique'
tff(fact_7233_of__int__round__abs__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)),X))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7234_of__int__round__abs__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7235_of__int__round__gt,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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_7236_of__int__round__gt,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))) ).

% of_int_round_gt
tff(fact_7237_of__int__round__ge,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))) ).

% of_int_round_ge
tff(fact_7238_of__int__round__ge,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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_7239_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_7240_round__0,axiom,
    archim7778729529865785530nd_rat(zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_7241_round__numeral,axiom,
    ! [Na: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(Na)) = aa_num_int(numeral_numeral_int,Na) ).

% round_numeral
tff(fact_7242_round__numeral,axiom,
    ! [Na: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(Na)) = aa_num_int(numeral_numeral_int,Na) ).

% round_numeral
tff(fact_7243_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_7244_round__1,axiom,
    archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).

% round_1
tff(fact_7245_round__neg__numeral,axiom,
    ! [Na: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% round_neg_numeral
tff(fact_7246_round__neg__numeral,axiom,
    ! [Na: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% round_neg_numeral
tff(fact_7247_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7778729529865785530nd_rat(X)),archim7778729529865785530nd_rat(Y)) ) ).

% round_mono
tff(fact_7248_round__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim8280529875227126926d_real,Y)) ) ).

% round_mono
tff(fact_7249_floor__le__round,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim8280529875227126926d_real,X)) ).

% floor_le_round
tff(fact_7250_floor__le__round,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),archim7778729529865785530nd_rat(X)) ).

% floor_le_round
tff(fact_7251_ceiling__ge__round,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim7802044766580827645g_real,X)) ).

% ceiling_ge_round
tff(fact_7252_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Z))))),abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,M)))) ).

% round_diff_minimal
tff(fact_7253_round__diff__minimal,axiom,
    ! [Z: real,M: int] : aa_real_o(aa_real_fun_real_o(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_7254_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),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_7255_round__def,axiom,
    ! [X: rat] : archim7778729529865785530nd_rat(X) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_7256_of__int__round__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7257_of__int__round__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7258_round__altdef,axiom,
    ! [X: real] :
      aa_real_int(archim8280529875227126926d_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)),aa_real_int(archim7802044766580827645g_real,X),aa_real_int(archim6058952711729229775r_real,X)) ).

% round_altdef
tff(fact_7259_round__altdef,axiom,
    ! [X: rat] :
      archim7778729529865785530nd_rat(X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),archimedean_frac_rat(X)),archim2889992004027027881ng_rat(X),archim3151403230148437115or_rat(X)) ).

% round_altdef
tff(fact_7260_cot__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_7261_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_7262_tan__cot_H,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_7263_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,X)) ) ) ).

% cot_gt_zero
tff(fact_7264_cot__zero,axiom,
    aa_complex_complex(cot_complex,zero_zero_complex) = zero_zero_complex ).

% cot_zero
tff(fact_7265_cot__zero,axiom,
    aa_real_real(cot_real,zero_zero_real) = zero_zero_real ).

% cot_zero
tff(fact_7266_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_7267_frac__of__int,axiom,
    ! [Z: int] : archimedean_frac_rat(aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_7268_frac__ge__0,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),archimedean_frac_rat(X)) ).

% frac_ge_0
tff(fact_7269_frac__ge__0,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X)) ).

% frac_ge_0
tff(fact_7270_frac__lt__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real) ).

% frac_lt_1
tff(fact_7271_frac__lt__1,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,archimedean_frac_rat(X)),one_one_rat) ).

% frac_lt_1
tff(fact_7272_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_7273_frac__1__eq,axiom,
    ! [X: rat] : archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = archimedean_frac_rat(X) ).

% frac_1_eq
tff(fact_7274_cot__def,axiom,
    ! [X3: complex] : aa_complex_complex(cot_complex,X3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,X3)),aa_complex_complex(sin_complex,X3)) ).

% cot_def
tff(fact_7275_cot__def,axiom,
    ! [X3: real] : aa_real_real(cot_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,X3)),aa_real_real(sin_real,X3)) ).

% cot_def
tff(fact_7276_frac__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat(X) = X )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% frac_eq
tff(fact_7277_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% frac_eq
tff(fact_7278_frac__add,axiom,
    ! [X: real,Y: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = $ite(aa_real_o(aa_real_fun_real_o(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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,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_7279_frac__add,axiom,
    ! [X: rat,Y: rat] :
      archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y)),minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y)),one_one_rat)) ).

% frac_add
tff(fact_7280_floor__add,axiom,
    ! [X: real,Y: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = $ite(aa_real_o(aa_real_fun_real_o(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_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,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_7281_floor__add,axiom,
    ! [X: rat,Y: rat] :
      archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y))),one_one_int)) ).

% floor_add
tff(fact_7282_powr__real__of__int,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(power_power_real(X),aa_int_nat(nat2,Na)),aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Na))))) ) ) ).

% powr_real_of_int
tff(fact_7283_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_7284_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_7285_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_7286_cmod__complex__polar,axiom,
    ! [R4: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R4)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R4) ).

% cmod_complex_polar
tff(fact_7287_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_7288_inverse__inverse__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ).

% inverse_inverse_eq
tff(fact_7289_inverse__inverse__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ).

% inverse_inverse_eq
tff(fact_7290_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7291_inverse__eq__iff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7292_inverse__eq__iff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_7293_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7294_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7295_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7296_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_7297_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_7298_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_7299_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_7300_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_7301_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).

% inverse_mult_distrib
tff(fact_7302_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(inverse_inverse_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_7303_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,X) = one_one_complex )
    <=> ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_7304_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,X) = one_one_rat )
    <=> ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_7305_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_7306_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_7307_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_7308_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) ).

% inverse_divide
tff(fact_7309_inverse__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(B),A) ).

% inverse_divide
tff(fact_7310_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,divide_divide_rat(A,B)) = divide_divide_rat(B,A) ).

% inverse_divide
tff(fact_7311_inverse__minus__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ).

% inverse_minus_eq
tff(fact_7312_inverse__minus__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ).

% inverse_minus_eq
tff(fact_7313_inverse__minus__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,uminus_uminus_rat(A)) = uminus_uminus_rat(aa_rat_rat(inverse_inverse_rat,A)) ).

% inverse_minus_eq
tff(fact_7314_abs__inverse,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ).

% abs_inverse
tff(fact_7315_abs__inverse,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(abs_abs_complex,A)) ).

% abs_inverse
tff(fact_7316_abs__inverse,axiom,
    ! [A: rat] : abs_abs_rat(aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,abs_abs_rat(A)) ).

% abs_inverse
tff(fact_7317_inverse__sgn,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% inverse_sgn
tff(fact_7318_inverse__sgn,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,sgn_sgn_rat(A)) = sgn_sgn_rat(A) ).

% inverse_sgn
tff(fact_7319_sgn__inverse,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) ).

% sgn_inverse
tff(fact_7320_sgn__inverse,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% sgn_inverse
tff(fact_7321_sgn__inverse,axiom,
    ! [A: rat] : sgn_sgn_rat(aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,sgn_sgn_rat(A)) ).

% sgn_inverse
tff(fact_7322_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7323_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7324_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7325_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7326_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_7327_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_7328_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_7329_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_7330_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7331_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7332_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_7333_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_7334_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_7335_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_7336_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_7337_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7338_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7339_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_7340_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_7341_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_7342_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_7343_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_7344_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_7345_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_7346_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_7347_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(W)) ).

% inverse_eq_divide_numeral
tff(fact_7348_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).

% inverse_eq_divide_numeral
tff(fact_7349_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,numeral_numeral_rat(W)) = divide_divide_rat(one_one_rat,numeral_numeral_rat(W)) ).

% inverse_eq_divide_numeral
tff(fact_7350_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_7351_divide__i,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).

% divide_i
tff(fact_7352_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_7353_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7354_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7355_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,uminus_uminus_rat(numeral_numeral_rat(W))) = divide_divide_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7356_divide__numeral__i,axiom,
    ! [Z: complex,Na: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Na)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Na)) ).

% divide_numeral_i
tff(fact_7357_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_7358_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_7359_i__even__power,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) ).

% i_even_power
tff(fact_7360_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_7361_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_7362_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_7363_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7364_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,X: complex] :
      ( ( aa_complex_complex(times_times_complex(Y),X) = aa_complex_complex(times_times_complex(X),Y) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7365_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( aa_rat_rat(times_times_rat(Y),X) = aa_rat_rat(times_times_rat(X),Y) )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Y)),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7366_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_7367_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_7368_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_7369_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_7370_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_7371_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_7372_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7373_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7374_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7375_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7376_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7377_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7378_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7379_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7380_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7381_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7382_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_7383_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7384_inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7385_inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_7386_power__inverse,axiom,
    ! [A: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,A)),Na) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(A),Na)) ).

% power_inverse
tff(fact_7387_power__inverse,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),Na) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),Na)) ).

% power_inverse
tff(fact_7388_power__inverse,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),Na) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),Na)) ).

% power_inverse
tff(fact_7389_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_7390_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_7391_norm__inverse__le__norm,axiom,
    ! [R4: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R4),aa_real_real(real_V7735802525324610683m_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,X))),aa_real_real(inverse_inverse_real,R4)) ) ) ).

% norm_inverse_le_norm
tff(fact_7392_norm__inverse__le__norm,axiom,
    ! [R4: real,X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R4),aa_complex_real(real_V1022390504157884413omplex,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,X))),aa_real_real(inverse_inverse_real,R4)) ) ) ).

% norm_inverse_le_norm
tff(fact_7393_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_7394_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_7395_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_7396_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_7397_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_7398_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_7399_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_7400_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_7401_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7402_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7403_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7404_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7405_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_7406_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_7407_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_7408_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_7409_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7410_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7411_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7412_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7413_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7414_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,uminus_uminus_rat(A)) = uminus_uminus_rat(aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7415_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,numeral_numeral_real(one)) = numeral_numeral_real(one) ).

% inverse_numeral_1
tff(fact_7416_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_7417_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,numeral_numeral_rat(one)) = numeral_numeral_rat(one) ).

% inverse_numeral_1
tff(fact_7418_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_7419_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).

% inverse_unique
tff(fact_7420_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).

% inverse_unique
tff(fact_7421_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_7422_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% field_class.field_divide_inverse
tff(fact_7423_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% field_class.field_divide_inverse
tff(fact_7424_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_7425_divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% divide_inverse
tff(fact_7426_divide__inverse,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% divide_inverse
tff(fact_7427_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_7428_divide__inverse__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).

% divide_inverse_commute
tff(fact_7429_divide__inverse__commute,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),A) ).

% divide_inverse_commute
tff(fact_7430_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ).

% inverse_eq_divide
tff(fact_7431_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_7432_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ).

% inverse_eq_divide
tff(fact_7433_power__mult__inverse__distrib,axiom,
    ! [X: real,M: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_real_real(inverse_inverse_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,X)),aa_nat_real(power_power_real(X),M)) ).

% power_mult_inverse_distrib
tff(fact_7434_power__mult__inverse__distrib,axiom,
    ! [X: complex,M: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(invers8013647133539491842omplex,X)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,X)),aa_nat_complex(power_power_complex(X),M)) ).

% power_mult_inverse_distrib
tff(fact_7435_power__mult__inverse__distrib,axiom,
    ! [X: rat,M: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(inverse_inverse_rat,X)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,X)),aa_nat_rat(power_power_rat(X),M)) ).

% power_mult_inverse_distrib
tff(fact_7436_power__mult__power__inverse__commute,axiom,
    ! [X: real,M: nat,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),Na)),aa_nat_real(power_power_real(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_7437_power__mult__power__inverse__commute,axiom,
    ! [X: complex,M: nat,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),Na)),aa_nat_complex(power_power_complex(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_7438_power__mult__power__inverse__commute,axiom,
    ! [X: rat,M: nat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),Na)),aa_nat_rat(power_power_rat(X),M)) ).

% power_mult_power_inverse_commute
tff(fact_7439_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa))) ).

% mult_inverse_of_nat_commute
tff(fact_7440_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa))) ).

% mult_inverse_of_nat_commute
tff(fact_7441_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa))),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa))) ).

% mult_inverse_of_nat_commute
tff(fact_7442_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7443_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,abs_abs_rat(A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7444_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_7445_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_7446_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_7447_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_7448_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_7449_cis__divide,axiom,
    ! [A: real,B: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% cis_divide
tff(fact_7450_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7451_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7452_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7453_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7454_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_7455_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_7456_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_7457_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_7458_inverse__le__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,X)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% inverse_le_1_iff
tff(fact_7459_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% inverse_le_1_iff
tff(fact_7460_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_7461_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_7462_one__less__inverse__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,X))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_7463_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_7464_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_7465_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_7466_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_7467_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7468_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7469_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7470_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_7471_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_7472_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_7473_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7474_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7475_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),minus_minus_rat(B,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7476_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7477_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7478_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7479_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_7480_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_7481_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_7482_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_7483_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_7484_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_7485_one__le__inverse__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,X))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_7486_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_7487_inverse__less__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,X)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% inverse_less_1_iff
tff(fact_7488_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% inverse_less_1_iff
tff(fact_7489_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_7490_one__le__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_le_inverse
tff(fact_7491_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = 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_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7492_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7493_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A),aa_rat_rat(inverse_inverse_rat,B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),minus_minus_rat(A,B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7494_reals__Archimedean,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),X) ) ).

% reals_Archimedean
tff(fact_7495_reals__Archimedean,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),X) ) ).

% reals_Archimedean
tff(fact_7496_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D3: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_7497_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D3: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_7498_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E2) ) ) ).

% real_arch_inverse
tff(fact_7499_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_7500_ln__inverse,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_7501_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))),X) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7502_ex__inverse__of__nat__less,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),X) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7503_power__diff__conv__inverse,axiom,
    ! [X: real,M: nat,Na: nat] :
      ( ( X != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),Na)),aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7504_power__diff__conv__inverse,axiom,
    ! [X: complex,M: nat,Na: nat] :
      ( ( X != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_complex(power_power_complex(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),Na)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7505_power__diff__conv__inverse,axiom,
    ! [X: rat,M: nat,Na: nat] :
      ( ( X != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_rat(power_power_rat(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Na)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),M)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7506_log__inverse,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(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_7507_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_7508_exp__plus__inverse__exp,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_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_7509_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R2: real,A4: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A4))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A4))))) ).

% complex_split_polar
tff(fact_7510_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_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_7511_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_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_7512_tan__cot,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_7513_real__le__x__sinh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_7514_real__le__abs__sinh,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_7515_tan__sec,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(cos_real,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_7516_tan__sec,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_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)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_7517_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_7518_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_7519_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_7520_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_7521_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)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_7522_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_7523_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_7524_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_7525_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_7526_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_7527_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_7528_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_7529_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_7530_cis__multiple__2pi,axiom,
    ! [Na: real] :
      ( member_real(Na,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Na)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_7531_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_7532_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% Suc_0_xor_eq
tff(fact_7533_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),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,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% xor_Suc_0_eq
tff(fact_7534_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_7535_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_7536_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_7537_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_7538_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_7539_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_7540_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_7541_bit_Oxor__self,axiom,
    ! [X: int] : aa_int_int(bit_se6526347334894502574or_int(X),X) = zero_zero_int ).

% bit.xor_self
tff(fact_7542_sinh__0,axiom,
    aa_complex_complex(sinh_complex,zero_zero_complex) = zero_zero_complex ).

% sinh_0
tff(fact_7543_sinh__0,axiom,
    aa_real_real(sinh_real,zero_zero_real) = zero_zero_real ).

% sinh_0
tff(fact_7544_take__bit__xor,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_se6526347334894502574or_int(A),B)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se2923211474154528505it_int(Na),A)),aa_int_int(bit_se2923211474154528505it_int(Na),B)) ).

% take_bit_xor
tff(fact_7545_take__bit__xor,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),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,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),B)) ).

% take_bit_xor
tff(fact_7546_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ).

% sinh_real_less_iff
tff(fact_7547_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_7548_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_7549_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_7550_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% sinh_real_pos_iff
tff(fact_7551_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% frac_eq_0_iff
tff(fact_7552_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat(X) = zero_zero_rat )
    <=> member_rat(X,ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_7553_floor__add2,axiom,
    ! [X: real,Y: real] :
      ( ( member_real(X,ring_1_Ints_real)
        | member_real(Y,ring_1_Ints_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)) ) ) ).

% floor_add2
tff(fact_7554_floor__add2,axiom,
    ! [X: rat,Y: rat] :
      ( ( member_rat(X,ring_1_Ints_rat)
        | member_rat(Y,ring_1_Ints_rat) )
     => ( archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y)) ) ) ).

% floor_add2
tff(fact_7555_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))
    <=> ~ member_real(X,ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_7556_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),archimedean_frac_rat(X))
    <=> ~ member_rat(X,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_7557_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_7558_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_7559_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_7560_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_7561_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_7562_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_7563_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_7564_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_7565_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_7566_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_7567_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_7568_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_7569_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_7570_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_7571_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_7572_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_7573_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_7574_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_7575_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_7576_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_7577_Ints__0,axiom,
    member_complex(zero_zero_complex,ring_1_Ints_complex) ).

% Ints_0
tff(fact_7578_Ints__0,axiom,
    member_real(zero_zero_real,ring_1_Ints_real) ).

% Ints_0
tff(fact_7579_Ints__0,axiom,
    member_rat(zero_zero_rat,ring_1_Ints_rat) ).

% Ints_0
tff(fact_7580_Ints__0,axiom,
    member_int(zero_zero_int,ring_1_Ints_int) ).

% Ints_0
tff(fact_7581_Ints__power,axiom,
    ! [A: int,Na: nat] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_nat_int(power_power_int(A),Na),ring_1_Ints_int) ) ).

% Ints_power
tff(fact_7582_Ints__power,axiom,
    ! [A: real,Na: nat] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_nat_real(power_power_real(A),Na),ring_1_Ints_real) ) ).

% Ints_power
tff(fact_7583_Ints__power,axiom,
    ! [A: complex,Na: nat] :
      ( member_complex(A,ring_1_Ints_complex)
     => member_complex(aa_nat_complex(power_power_complex(A),Na),ring_1_Ints_complex) ) ).

% Ints_power
tff(fact_7584_Ints__power,axiom,
    ! [A: rat,Na: nat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(aa_nat_rat(power_power_rat(A),Na),ring_1_Ints_rat) ) ).

% Ints_power
tff(fact_7585_Ints__of__nat,axiom,
    ! [Na: nat] : member_real(aa_nat_real(semiri5074537144036343181t_real,Na),ring_1_Ints_real) ).

% Ints_of_nat
tff(fact_7586_Ints__of__nat,axiom,
    ! [Na: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,Na),ring_1_Ints_int) ).

% Ints_of_nat
tff(fact_7587_Ints__of__nat,axiom,
    ! [Na: nat] : member_complex(aa_nat_complex(semiri8010041392384452111omplex,Na),ring_1_Ints_complex) ).

% Ints_of_nat
tff(fact_7588_Ints__of__nat,axiom,
    ! [Na: nat] : member_rat(aa_nat_rat(semiri681578069525770553at_rat,Na),ring_1_Ints_rat) ).

% Ints_of_nat
tff(fact_7589_bit_Oconj__xor__distrib,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(bit_se725231765392027082nd_int(X),aa_int_int(bit_se6526347334894502574or_int(Y),Z)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se725231765392027082nd_int(X),Y)),aa_int_int(bit_se725231765392027082nd_int(X),Z)) ).

% bit.conj_xor_distrib
tff(fact_7590_bit_Oconj__xor__distrib2,axiom,
    ! [Y: int,Z: int,X: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se6526347334894502574or_int(Y),Z)),X) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se725231765392027082nd_int(Y),X)),aa_int_int(bit_se725231765392027082nd_int(Z),X)) ).

% bit.conj_xor_distrib2
tff(fact_7591_bit__xor__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)),Na)
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
        <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Na) ) ) ).

% bit_xor_iff
tff(fact_7592_bit__xor__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(A),B)),Na)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
        <=> aa_nat_o(bit_se1146084159140164899it_int(B),Na) ) ) ).

% bit_xor_iff
tff(fact_7593_Ints__abs,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_int_int(abs_abs_int,A),ring_1_Ints_int) ) ).

% Ints_abs
tff(fact_7594_Ints__abs,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(abs_abs_Code_integer(A),ring_11222124179247155820nteger) ) ).

% Ints_abs
tff(fact_7595_Ints__abs,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(abs_abs_rat(A),ring_1_Ints_rat) ) ).

% Ints_abs
tff(fact_7596_Ints__abs,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_real_real(abs_abs_real,A),ring_1_Ints_real) ) ).

% Ints_abs
tff(fact_7597_cosh__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y))) ).

% cosh_add
tff(fact_7598_sinh__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,X)),aa_real_real(sinh_real,Y))) ).

% sinh_add
tff(fact_7599_sinh__less__cosh__real,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) ).

% sinh_less_cosh_real
tff(fact_7600_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_7601_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_7602_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_7603_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_7604_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_7605_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_7606_of__nat__xor__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_xor_eq
tff(fact_7607_of__nat__xor__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na)) = aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_xor_eq
tff(fact_7608_Ints__diff,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(minus_minus_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_diff
tff(fact_7609_Ints__diff,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_diff
tff(fact_7610_Ints__diff,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(minus_minus_rat(A,B),ring_1_Ints_rat) ) ) ).

% Ints_diff
tff(fact_7611_Ints__diff,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_diff
tff(fact_7612_Ints__add,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(plus_plus_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_add
tff(fact_7613_Ints__add,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_add
tff(fact_7614_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_7615_Ints__add,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_add
tff(fact_7616_Ints__1,axiom,
    member_complex(one_one_complex,ring_1_Ints_complex) ).

% Ints_1
tff(fact_7617_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_7618_Ints__1,axiom,
    member_int(one_one_int,ring_1_Ints_int) ).

% Ints_1
tff(fact_7619_Ints__1,axiom,
    member_real(one_one_real,ring_1_Ints_real) ).

% Ints_1
tff(fact_7620_Ints__mult,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(times_times_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_mult
tff(fact_7621_Ints__mult,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_mult
tff(fact_7622_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(times_times_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_7623_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_mult
tff(fact_7624_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_7625_tanh__def,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,X)) ).

% tanh_def
tff(fact_7626_tanh__def,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) ).

% tanh_def
tff(fact_7627_Ints__minus,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_real_real(uminus_uminus_real,A),ring_1_Ints_real) ) ).

% Ints_minus
tff(fact_7628_Ints__minus,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_int_int(uminus_uminus_int,A),ring_1_Ints_int) ) ).

% Ints_minus
tff(fact_7629_Ints__minus,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => member_complex(aa_complex_complex(uminus1482373934393186551omplex,A),ring_1_Ints_complex) ) ).

% Ints_minus
tff(fact_7630_Ints__minus,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),ring_11222124179247155820nteger) ) ).

% Ints_minus
tff(fact_7631_Ints__minus,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(uminus_uminus_rat(A),ring_1_Ints_rat) ) ).

% Ints_minus
tff(fact_7632_minus__in__Ints__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(uminus_uminus_real,X),ring_1_Ints_real)
    <=> member_real(X,ring_1_Ints_real) ) ).

% minus_in_Ints_iff
tff(fact_7633_minus__in__Ints__iff,axiom,
    ! [X: int] :
      ( member_int(aa_int_int(uminus_uminus_int,X),ring_1_Ints_int)
    <=> member_int(X,ring_1_Ints_int) ) ).

% minus_in_Ints_iff
tff(fact_7634_minus__in__Ints__iff,axiom,
    ! [X: complex] :
      ( member_complex(aa_complex_complex(uminus1482373934393186551omplex,X),ring_1_Ints_complex)
    <=> member_complex(X,ring_1_Ints_complex) ) ).

% minus_in_Ints_iff
tff(fact_7635_minus__in__Ints__iff,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),ring_11222124179247155820nteger)
    <=> member_Code_integer(X,ring_11222124179247155820nteger) ) ).

% minus_in_Ints_iff
tff(fact_7636_minus__in__Ints__iff,axiom,
    ! [X: rat] :
      ( member_rat(uminus_uminus_rat(X),ring_1_Ints_rat)
    <=> member_rat(X,ring_1_Ints_rat) ) ).

% minus_in_Ints_iff
tff(fact_7637_Ints__numeral,axiom,
    ! [Na: num] : member_complex(numera6690914467698888265omplex(Na),ring_1_Ints_complex) ).

% Ints_numeral
tff(fact_7638_Ints__numeral,axiom,
    ! [Na: num] : member_real(numeral_numeral_real(Na),ring_1_Ints_real) ).

% Ints_numeral
tff(fact_7639_Ints__numeral,axiom,
    ! [Na: num] : member_rat(numeral_numeral_rat(Na),ring_1_Ints_rat) ).

% Ints_numeral
tff(fact_7640_Ints__numeral,axiom,
    ! [Na: num] : member_int(aa_num_int(numeral_numeral_int,Na),ring_1_Ints_int) ).

% Ints_numeral
tff(fact_7641_cosh__plus__sinh,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cosh_complex,X)),aa_complex_complex(sinh_complex,X)) = aa_complex_complex(exp_complex,X) ).

% cosh_plus_sinh
tff(fact_7642_cosh__plus__sinh,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cosh_real,X)),aa_real_real(sinh_real,X)) = aa_real_real(exp_real,X) ).

% cosh_plus_sinh
tff(fact_7643_sinh__plus__cosh,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,X)) = aa_complex_complex(exp_complex,X) ).

% sinh_plus_cosh
tff(fact_7644_sinh__plus__cosh,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) = aa_real_real(exp_real,X) ).

% sinh_plus_cosh
tff(fact_7645_Ints__of__int,axiom,
    ! [Z: int] : member_complex(aa_int_complex(ring_17405671764205052669omplex,Z),ring_1_Ints_complex) ).

% Ints_of_int
tff(fact_7646_Ints__of__int,axiom,
    ! [Z: int] : member_int(aa_int_int(ring_1_of_int_int,Z),ring_1_Ints_int) ).

% Ints_of_int
tff(fact_7647_Ints__of__int,axiom,
    ! [Z: int] : member_real(aa_int_real(ring_1_of_int_real,Z),ring_1_Ints_real) ).

% Ints_of_int
tff(fact_7648_Ints__of__int,axiom,
    ! [Z: int] : member_rat(aa_int_rat(ring_1_of_int_rat,Z),ring_1_Ints_rat) ).

% Ints_of_int
tff(fact_7649_Ints__induct,axiom,
    ! [Q3: complex,P: fun_complex_o] :
      ( member_complex(Q3,ring_1_Ints_complex)
     => ( ! [Z2: int] : aa_complex_o(P,aa_int_complex(ring_17405671764205052669omplex,Z2))
       => aa_complex_o(P,Q3) ) ) ).

% Ints_induct
tff(fact_7650_Ints__induct,axiom,
    ! [Q3: int,P: fun_int_o] :
      ( member_int(Q3,ring_1_Ints_int)
     => ( ! [Z2: int] : aa_int_o(P,aa_int_int(ring_1_of_int_int,Z2))
       => aa_int_o(P,Q3) ) ) ).

% Ints_induct
tff(fact_7651_Ints__induct,axiom,
    ! [Q3: real,P: fun_real_o] :
      ( member_real(Q3,ring_1_Ints_real)
     => ( ! [Z2: int] : aa_real_o(P,aa_int_real(ring_1_of_int_real,Z2))
       => aa_real_o(P,Q3) ) ) ).

% Ints_induct
tff(fact_7652_Ints__induct,axiom,
    ! [Q3: rat,P: fun_rat_o] :
      ( member_rat(Q3,ring_1_Ints_rat)
     => ( ! [Z2: int] : aa_rat_o(P,aa_int_rat(ring_1_of_int_rat,Z2))
       => aa_rat_o(P,Q3) ) ) ).

% Ints_induct
tff(fact_7653_Ints__cases,axiom,
    ! [Q3: complex] :
      ( member_complex(Q3,ring_1_Ints_complex)
     => ~ ! [Z2: int] : Q3 != aa_int_complex(ring_17405671764205052669omplex,Z2) ) ).

% Ints_cases
tff(fact_7654_Ints__cases,axiom,
    ! [Q3: int] :
      ( member_int(Q3,ring_1_Ints_int)
     => ~ ! [Z2: int] : Q3 != aa_int_int(ring_1_of_int_int,Z2) ) ).

% Ints_cases
tff(fact_7655_Ints__cases,axiom,
    ! [Q3: real] :
      ( member_real(Q3,ring_1_Ints_real)
     => ~ ! [Z2: int] : Q3 != aa_int_real(ring_1_of_int_real,Z2) ) ).

% Ints_cases
tff(fact_7656_Ints__cases,axiom,
    ! [Q3: rat] :
      ( member_rat(Q3,ring_1_Ints_rat)
     => ~ ! [Z2: int] : Q3 != aa_int_rat(ring_1_of_int_rat,Z2) ) ).

% Ints_cases
tff(fact_7657_cosh__real__pos,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X)) ).

% cosh_real_pos
tff(fact_7658_cosh__real__ge__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X)) ).

% cosh_real_ge_1
tff(fact_7659_sinh__double,axiom,
    ! [X: complex] : aa_complex_complex(sinh_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(sinh_complex,X))),aa_complex_complex(cosh_complex,X)) ).

% sinh_double
tff(fact_7660_sinh__double,axiom,
    ! [X: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sinh_real,X))),aa_real_real(cosh_real,X)) ).

% sinh_double
tff(fact_7661_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( ( aa_complex_complex(plus_plus_complex(A),A) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7662_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7663_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7664_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7665_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_7666_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_7667_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_7668_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A)),A) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_7669_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_7670_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_7671_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_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),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_7672_cosh__square__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% cosh_square_eq
tff(fact_7673_cosh__square__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% cosh_square_eq
tff(fact_7674_hyperbolic__pythagoras,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% hyperbolic_pythagoras
tff(fact_7675_hyperbolic__pythagoras,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% hyperbolic_pythagoras
tff(fact_7676_sinh__square__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% sinh_square_eq
tff(fact_7677_sinh__square__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% sinh_square_eq
tff(fact_7678_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_complex(aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(ring_17405671764205052669omplex,A)),aa_int_complex(ring_17405671764205052669omplex,B)),ring_1_Ints_complex) ) ).

% of_int_divide_in_Ints
tff(fact_7679_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_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)),ring_1_Ints_real) ) ).

% of_int_divide_in_Ints
tff(fact_7680_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B)),ring_1_Ints_rat) ) ).

% of_int_divide_in_Ints
tff(fact_7681_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(ring_1_of_int_int,A)),aa_int_int(ring_1_of_int_int,B)),ring_1_Ints_int) ) ).

% of_int_divide_in_Ints
tff(fact_7682_cosh__double,axiom,
    ! [X: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_7683_cosh__double,axiom,
    ! [X: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_7684_even__xor__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3222712562003087583nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_xor_iff
tff(fact_7685_even__xor__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_xor_iff
tff(fact_7686_even__xor__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_xor_iff
tff(fact_7687_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% Ints_odd_less_0
tff(fact_7688_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_7689_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(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),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_7690_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7691_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( ( X != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),abs_abs_rat(X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7692_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( ( X != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7693_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( ( X != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7694_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7695_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7696_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(X)),one_one_rat)
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7697_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7698_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y,ring_11222124179247155820nteger)
       => ( ( X = Y )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,Y))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7699_Ints__eq__abs__less1,axiom,
    ! [X: real,Y: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( member_real(Y,ring_1_Ints_real)
       => ( ( X = Y )
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),one_one_real) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7700_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( member_rat(Y,ring_1_Ints_rat)
       => ( ( X = Y )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(X,Y))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7701_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( member_int(Y,ring_1_Ints_int)
       => ( ( X = Y )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7702_frac__neg,axiom,
    ! [X: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = $ite(member_real(X,ring_1_Ints_real),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_neg
tff(fact_7703_frac__neg,axiom,
    ! [X: rat] :
      archimedean_frac_rat(uminus_uminus_rat(X)) = $ite(member_rat(X,ring_1_Ints_rat),zero_zero_rat,minus_minus_rat(one_one_rat,archimedean_frac_rat(X))) ).

% frac_neg
tff(fact_7704_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_rat(ring_1_of_int_rat,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7705_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_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_int_int(ring_1_of_int_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7706_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7707_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7708_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_int(ring_1_of_int_int,archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7709_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_real(ring_1_of_int_real,archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7710_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( archimedean_frac_rat(X) = A )
    <=> ( member_rat(minus_minus_rat(X,A),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_7711_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = A )
    <=> ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A),ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) ) ) ).

% frac_unique_iff
tff(fact_7712_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7713_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7714_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7715_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_rat(ring_1_of_int_rat,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_Ints
tff(fact_7716_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_int(ring_1_of_int_int,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_Ints
tff(fact_7717_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))),aa_int_real(ring_1_of_int_real,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_Ints
tff(fact_7718_tanh__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cosh_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y)))) ) ) ) ).

% tanh_add
tff(fact_7719_tanh__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cosh_real,X) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y) != zero_zero_real )
       => ( aa_real_real(tanh_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(tanh_real,X)),aa_real_real(tanh_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(tanh_real,X)),aa_real_real(tanh_real,Y)))) ) ) ) ).

% tanh_add
tff(fact_7720_sin__integer__2pi,axiom,
    ! [Na: real] :
      ( member_real(Na,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Na)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_7721_cos__integer__2pi,axiom,
    ! [Na: real] :
      ( member_real(Na,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Na)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_7722_cosh__field__def,axiom,
    ! [Z: real] : aa_real_real(cosh_real,Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_7723_cosh__field__def,axiom,
    ! [Z: complex] : aa_complex_complex(cosh_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Z)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_7724_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,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,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_7725_xor__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_7726_xor__one__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(A,one_one_Code_integer) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_7727_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_7728_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_7729_one__xor__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(one_one_Code_integer,A) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_7730_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_7731_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_7732_sinh__field__def,axiom,
    ! [Z: real] : aa_real_real(sinh_real,Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_7733_sinh__field__def,axiom,
    ! [Z: complex] : aa_complex_complex(sinh_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Z)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_7734_cosh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cosh_real,X) = zero_zero_real )
    <=> ( aa_nat_real(power_power_real(aa_real_real(exp_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_7735_cosh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cosh_complex,X) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_7736_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7737_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7738_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7739_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_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_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_7740_push__bit__numeral__minus__1,axiom,
    ! [Na: num] : bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,Na),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Na))) ).

% push_bit_numeral_minus_1
tff(fact_7741_push__bit__numeral__minus__1,axiom,
    ! [Na: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,Na)),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))),aa_num_nat(numeral_numeral_nat,Na))) ).

% push_bit_numeral_minus_1
tff(fact_7742_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ( A2 != B2 )
       => aa_set_nat_o(ord_less_set_nat(A2),B2) ) ) ).

% psubsetI
tff(fact_7743_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(X),Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xa = zero_zero_nat,
                (A4),
                $ite(Xa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
                   => $ite(
                        Xa = Mi,
                        $true,
                        $ite(
                          Xa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xa),
                              $false,
                              $let(
                                h: nat,
                                h:= 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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_7744_set__vebt_H__def,axiom,
    ! [Tb: vEBT_VEBT] : vEBT_VEBT_set_vebt(Tb) = collect_nat(vEBT_vebt_member(Tb)) ).

% set_vebt'_def
tff(fact_7745_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_7746_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_7747_push__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(bit_se545348938243370406it_int(Na),zero_zero_int) = zero_zero_int ).

% push_bit_of_0
tff(fact_7748_push__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% push_bit_of_0
tff(fact_7749_push__bit__eq__0__iff,axiom,
    ! [Na: nat,A: int] :
      ( ( aa_int_int(bit_se545348938243370406it_int(Na),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
tff(fact_7750_push__bit__eq__0__iff,axiom,
    ! [Na: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
tff(fact_7751_push__bit__push__bit,axiom,
    ! [M: nat,Na: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(M),aa_int_int(bit_se545348938243370406it_int(Na),A)) = aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),A) ).

% push_bit_push_bit
tff(fact_7752_push__bit__push__bit,axiom,
    ! [M: nat,Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),A) ).

% push_bit_push_bit
tff(fact_7753_push__bit__and,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se545348938243370406it_int(Na),A)),aa_int_int(bit_se545348938243370406it_int(Na),B)) ).

% push_bit_and
tff(fact_7754_push__bit__and,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),B)) ).

% push_bit_and
tff(fact_7755_push__bit__xor,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(bit_se6526347334894502574or_int(A),B)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se545348938243370406it_int(Na),A)),aa_int_int(bit_se545348938243370406it_int(Na),B)) ).

% push_bit_xor
tff(fact_7756_push__bit__xor,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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_se547839408752420682it_nat,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),B)) ).

% push_bit_xor
tff(fact_7757_concat__bit__of__zero__1,axiom,
    ! [Na: nat,L: int] : aa_int_int(bit_concat_bit(Na,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(Na),L) ).

% concat_bit_of_zero_1
tff(fact_7758_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_7759_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_7760_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_7761_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_7762_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_7763_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_7764_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(gbinomial_real(A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_7765_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_7766_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_7767_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_7768_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_7769_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_7770_push__bit__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,K)) = aa_int_int(bit_se545348938243370406it_int(Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) ).

% push_bit_Suc_numeral
tff(fact_7771_push__bit__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% push_bit_Suc_numeral
tff(fact_7772_push__bit__Suc__minus__numeral,axiom,
    ! [Na: nat,K: num] : bit_se7788150548672797655nteger(aa_nat_nat(suc,Na),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = bit_se7788150548672797655nteger(Na,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_7773_push__bit__Suc__minus__numeral,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_7774_push__bit__numeral,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,K)) = aa_int_int(bit_se545348938243370406it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) ).

% push_bit_numeral
tff(fact_7775_push__bit__numeral,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% push_bit_numeral
tff(fact_7776_push__bit__Suc,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Na)),A) = aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% push_bit_Suc
tff(fact_7777_push__bit__Suc,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,Na)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% push_bit_Suc
tff(fact_7778_push__bit__of__1,axiom,
    ! [Na: nat] : aa_int_int(bit_se545348938243370406it_int(Na),one_one_int) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) ).

% push_bit_of_1
tff(fact_7779_push__bit__of__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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))),Na) ).

% push_bit_of_1
tff(fact_7780_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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))),Na) ).

% push_bit_of_Suc_0
tff(fact_7781_even__push__bit__iff,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se7788150548672797655nteger(Na,A))
    <=> ( ( Na != zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_7782_even__push__bit__iff,axiom,
    ! [Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se545348938243370406it_int(Na),A))
    <=> ( ( Na != zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_7783_even__push__bit__iff,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A))
    <=> ( ( Na != zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_push_bit_iff
tff(fact_7784_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,L),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = bit_se7788150548672797655nteger(aa_num_nat(pred_numeral,L),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,K)))) ).

% push_bit_minus_numeral
tff(fact_7785_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_se545348938243370406it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ).

% push_bit_minus_numeral
tff(fact_7786_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(K),L)),Na)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Na) ) ) ).

% bit_xor_int_iff
tff(fact_7787_flip__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(Na),K) = aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(bit_se545348938243370406it_int(Na),one_one_int)) ).

% flip_bit_int_def
tff(fact_7788_push__bit__of__int,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se545348938243370406it_int(Na),K)) ).

% push_bit_of_int
tff(fact_7789_push__bit__add,axiom,
    ! [Na: nat,A: int,B: int] : aa_int_int(bit_se545348938243370406it_int(Na),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(bit_se545348938243370406it_int(Na),A)),aa_int_int(bit_se545348938243370406it_int(Na),B)) ).

% push_bit_add
tff(fact_7790_push__bit__add,axiom,
    ! [Na: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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(bit_se547839408752420682it_nat,Na),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),B)) ).

% push_bit_add
tff(fact_7791_push__bit__minus,axiom,
    ! [Na: nat,A: code_integer] : bit_se7788150548672797655nteger(Na,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_se7788150548672797655nteger(Na,A)) ).

% push_bit_minus
tff(fact_7792_push__bit__minus,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se545348938243370406it_int(Na),A)) ).

% push_bit_minus
tff(fact_7793_push__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(Na),K)) ).

% push_bit_nat_eq
tff(fact_7794_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_af,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_af,B)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% subset_divisors_dvd
tff(fact_7795_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ag,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_ag,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_7796_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ah,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ah,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_7797_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_7798_push__bit__of__nat,axiom,
    ! [Na: nat,M: nat] : aa_int_int(bit_se545348938243370406it_int(Na),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),M)) ).

% push_bit_of_nat
tff(fact_7799_push__bit__of__nat,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),M)) ).

% push_bit_of_nat
tff(fact_7800_of__nat__push__bit,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Na)) = aa_int_int(bit_se545348938243370406it_int(M),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_push_bit
tff(fact_7801_of__nat__push__bit,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_push_bit
tff(fact_7802_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_af,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_af,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7803_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7804_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ag,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_ag,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7805_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ah,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ah,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7806_less__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ord_less_set_complex(A2,B2)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_7807_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ord_less_set_real(A2,B2)
    <=> ord_less_fun_real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_7808_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ord_less_set_set_nat(A2,B2)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_7809_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_7810_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ord_less_set_int(A2,B2)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_7811_psubsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( ord_less_set_complex(A2,B2)
     => ( member_complex(C,A2)
       => member_complex(C,B2) ) ) ).

% psubsetD
tff(fact_7812_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ord_less_set_real(A2,B2)
     => ( member_real(C,A2)
       => member_real(C,B2) ) ) ).

% psubsetD
tff(fact_7813_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ord_less_set_set_nat(A2,B2)
     => ( member_set_nat(C,A2)
       => member_set_nat(C,B2) ) ) ).

% psubsetD
tff(fact_7814_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% psubsetD
tff(fact_7815_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ord_less_set_int(A2,B2)
     => ( member_int(C,A2)
       => member_int(C,B2) ) ) ).

% psubsetD
tff(fact_7816_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ord_less_set_complex(A2,B2)
     => ? [B4: complex] : member_complex(B4,minus_811609699411566653omplex(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_7817_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ord_less_set_real(A2,B2)
     => ? [B4: real] : member_real(B4,minus_minus_set_real(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_7818_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ord_less_set_set_nat(A2,B2)
     => ? [B4: set_nat] : member_set_nat(B4,minus_2163939370556025621et_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_7819_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ord_less_set_int(A2,B2)
     => ? [B4: int] : member_int(B4,minus_minus_set_int(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_7820_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ? [B4: nat] : aa_set_nat_o(member_nat(B4),minus_minus_set_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_7821_lambda__one,axiom,
    aTP_Lamm_aj = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_7822_lambda__one,axiom,
    aTP_Lamm_ak = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_7823_lambda__one,axiom,
    aTP_Lamm_al = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_7824_lambda__one,axiom,
    aTP_Lamm_am = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_7825_lambda__one,axiom,
    aTP_Lamm_an = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_7826_lambda__zero,axiom,
    aTP_Lamm_ao = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_7827_lambda__zero,axiom,
    aTP_Lamm_ap = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_7828_lambda__zero,axiom,
    aTP_Lamm_aq = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_7829_lambda__zero,axiom,
    aTP_Lamm_ar = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_7830_lambda__zero,axiom,
    aTP_Lamm_as = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_7831_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Na)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Na),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_7832_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numeral_numeral_real(aa_num_num(bit0,Na)) = $let(
        m2: real,
        m2:= numeral_numeral_real(Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).

% numeral_code(2)
tff(fact_7833_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numeral_numeral_rat(aa_num_num(bit0,Na)) = $let(
        m2: rat,
        m2:= numeral_numeral_rat(Na),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2) ) ).

% numeral_code(2)
tff(fact_7834_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_7835_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_7836_nat__less__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_less_as_int
tff(fact_7837_nat__leq__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_leq_as_int
tff(fact_7838_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Na)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Na),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_7839_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numeral_numeral_real(aa_num_num(bit1,Na)) = $let(
        m2: real,
        m2:= numeral_numeral_real(Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_7840_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numeral_numeral_rat(aa_num_num(bit1,Na)) = $let(
        m2: rat,
        m2:= numeral_numeral_rat(Na),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_7841_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_7842_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_7843_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))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_7844_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_7845_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))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_7846_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))) = $let(
        w: nat,
        w:= 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(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_7847_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))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_7848_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))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7849_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(power_power_real(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7850_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))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7851_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))) = $let(
        w: nat,
        w:= 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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7852_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))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7853_nat__plus__as__int,axiom,
    ! [X3: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_7854_nat__div__as__int,axiom,
    ! [X3: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_7855_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(X),Y)) ) ) ).

% XOR_lower
tff(fact_7856_signed__take__bit__code,axiom,
    ! [Na: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Na,A) = $let(
        l: code_integer,
        l:= bit_se1745604003318907178nteger(aa_nat_nat(suc,Na),A),
        $ite(aa_nat_o(bit_se9216721137139052372nteger(l),Na),plus_p5714425477246183910nteger(l,bit_se7788150548672797655nteger(aa_nat_nat(suc,Na),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))),l) ) ).

% signed_take_bit_code
tff(fact_7857_signed__take__bit__code,axiom,
    ! [Na: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Na),A) = $let(
        l: int,
        l:= aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na)),A),
        $ite(aa_nat_o(bit_se1146084159140164899it_int(l),Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,l),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,one_one_int))),l) ) ).

% signed_take_bit_code
tff(fact_7858_set__conv__nth,axiom,
    ! [Xs: list_real] : set_real2(Xs) = collect_real(aTP_Lamm_at(Xs)) ).

% set_conv_nth
tff(fact_7859_set__conv__nth,axiom,
    ! [Xs: list_list_nat] : set_list_nat2(Xs) = collect_list_nat(aTP_Lamm_au(Xs)) ).

% set_conv_nth
tff(fact_7860_set__conv__nth,axiom,
    ! [Xs: list_set_nat] : set_set_nat2(Xs) = collect_set_nat(aTP_Lamm_av(Xs)) ).

% set_conv_nth
tff(fact_7861_set__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs) = collect_VEBT_VEBT(aTP_Lamm_aw(Xs)) ).

% set_conv_nth
tff(fact_7862_set__conv__nth,axiom,
    ! [Xs: list_o] : set_o2(Xs) = collect_o(aTP_Lamm_ax(Xs)) ).

% set_conv_nth
tff(fact_7863_set__conv__nth,axiom,
    ! [Xs: list_nat] : set_nat2(Xs) = collect_nat(aTP_Lamm_ay(Xs)) ).

% set_conv_nth
tff(fact_7864_set__conv__nth,axiom,
    ! [Xs: list_int] : set_int2(Xs) = collect_int(aTP_Lamm_az(Xs)) ).

% set_conv_nth
tff(fact_7865_push__bit__take__bit,axiom,
    ! [M: nat,Na: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(M),aa_int_int(bit_se2923211474154528505it_int(Na),A)) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_int_int(bit_se545348938243370406it_int(M),A)) ).

% push_bit_take_bit
tff(fact_7866_push__bit__take__bit,axiom,
    ! [M: nat,Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),A)) ).

% push_bit_take_bit
tff(fact_7867_take__bit__push__bit,axiom,
    ! [M: nat,Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_se545348938243370406it_int(Na),A)) = aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),A)) ).

% take_bit_push_bit
tff(fact_7868_take__bit__push__bit,axiom,
    ! [M: nat,Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),A)) ).

% take_bit_push_bit
tff(fact_7869_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_7870_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z4),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_7871_gbinomial__Suc__Suc,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(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7872_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_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(plus_plus_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7873_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7874_gbinomial__code,axiom,
    ! [A: complex,K: nat] :
      aa_nat_complex(gbinomial_complex(A),K) = $ite(K = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_ba(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K))) ).

% gbinomial_code
tff(fact_7875_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,divide_divide_rat(set_fo1949268297981939178at_rat(aTP_Lamm_bb(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat),one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,K))) ).

% gbinomial_code
tff(fact_7876_gbinomial__code,axiom,
    ! [A: real,K: nat] :
      aa_nat_real(gbinomial_real(A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_bc(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K))) ).

% gbinomial_code
tff(fact_7877_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Na)),K) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7878_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),K) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7879_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7880_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(M),K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_7881_xor__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% xor_nat_def
tff(fact_7882_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bd,X)) ).

% set_decode_def
tff(fact_7883_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
         => aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% psubsetE
tff(fact_7884_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_7885_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% psubset_imp_subset
tff(fact_7886_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C4)
       => aa_set_nat_o(ord_less_set_nat(A2),C4) ) ) ).

% psubset_subset_trans
tff(fact_7887_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% subset_not_subset_eq
tff(fact_7888_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_set_nat(B2),C4)
       => aa_set_nat_o(ord_less_set_nat(A2),C4) ) ) ).

% subset_psubset_trans
tff(fact_7889_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_set_nat(A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_7890_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q3)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_7891_concat__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Na,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_int_int(bit_se545348938243370406it_int(Na),L)) ).

% concat_bit_eq
tff(fact_7892_flip__bit__eq__xor,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(Na),A) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se545348938243370406it_int(Na),one_one_int)) ).

% flip_bit_eq_xor
tff(fact_7893_flip__bit__eq__xor,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Na),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),one_one_nat)) ).

% flip_bit_eq_xor
tff(fact_7894_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_7895_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_7896_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_7897_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(gbinomial_real(A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_7898_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_7899_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_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(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_7900_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_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_7901_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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(gbinomial_real(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7902_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_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(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(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7903_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_7904_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(aa_rat_fun_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_7905_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(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(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7906_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_7907_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(aa_rat_fun_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_7908_push__bit__double,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(aa_int_fun_int_int(times_times_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_int_int(bit_se545348938243370406it_int(Na),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% push_bit_double
tff(fact_7909_push__bit__double,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_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_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% push_bit_double
tff(fact_7910_pochhammer__code,axiom,
    ! [A: real,Na: nat] :
      aa_nat_real(comm_s7457072308508201937r_real(A),Na) = $ite(Na = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_be(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_7911_pochhammer__code,axiom,
    ! [A: int,Na: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Na) = $ite(Na = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_bf(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_7912_pochhammer__code,axiom,
    ! [A: complex,Na: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = $ite(Na = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_bg(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_7913_pochhammer__code,axiom,
    ! [A: rat,Na: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = $ite(Na = zero_zero_nat,one_one_rat,set_fo1949268297981939178at_rat(aTP_Lamm_bh(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),one_one_rat)) ).

% pochhammer_code
tff(fact_7914_pochhammer__code,axiom,
    ! [A: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Na) = $ite(Na = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_bi(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_7915_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Na)
    <=> ( aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se545348938243370406it_int(Na),one_one_int)) != zero_zero_int ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7916_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Na)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),one_one_nat)) != zero_zero_nat ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7917_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(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(gbinomial_real(A),K)) ).

% Suc_times_gbinomial
tff(fact_7918_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_7919_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(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_7920_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(gbinomial_real(A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_7921_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_7922_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(minus_minus_rat(A,one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_7923_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),M)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_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_7924_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_7925_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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(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_7926_push__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Na),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))),Na)) ).

% push_bit_int_def
tff(fact_7927_push__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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))),Na)) ).

% push_bit_nat_def
tff(fact_7928_push__bit__eq__mult,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Na),A) = 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))),Na)) ).

% push_bit_eq_mult
tff(fact_7929_push__bit__eq__mult,axiom,
    ! [Na: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),A) = 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))),Na)) ).

% push_bit_eq_mult
tff(fact_7930_exp__dvdE,axiom,
    ! [Na: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)),A)
     => ~ ! [B4: code_integer] : A != bit_se7788150548672797655nteger(Na,B4) ) ).

% exp_dvdE
tff(fact_7931_exp__dvdE,axiom,
    ! [Na: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),A)
     => ~ ! [B4: int] : A != aa_int_int(bit_se545348938243370406it_int(Na),B4) ) ).

% exp_dvdE
tff(fact_7932_exp__dvdE,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na)),A)
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),B4) ) ).

% exp_dvdE
tff(fact_7933_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S)),X)
    <=> $let(
          pos: nat,
          pos:= 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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_7934_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_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(gbinomial_real(A),K)) ).

% gbinomial_factors
tff(fact_7935_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_7936_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_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_7937_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_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(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_7938_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_7939_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_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)),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7940_gbinomial__index__swap,axiom,
    ! [K: nat,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_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)),Na)) ).

% gbinomial_index_swap
tff(fact_7941_gbinomial__index__swap,axiom,
    ! [K: nat,Na: 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,Na))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),Na)) ).

% gbinomial_index_swap
tff(fact_7942_gbinomial__index__swap,axiom,
    ! [K: nat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Na)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),Na)) ).

% gbinomial_index_swap
tff(fact_7943_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_7944_gbinomial__negated__upper,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = 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(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_7945_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(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K),A),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_7946_push__bit__minus__one,axiom,
    ! [Na: nat] : aa_int_int(bit_se545348938243370406it_int(Na),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))),Na)) ).

% push_bit_minus_one
tff(fact_7947_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd2)),X)
    <=> $let(
          pos: nat,
          pos:= 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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_7948_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi3: nat,Ma: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,V),TreeLista,Vc)),X)
    <=> ( ( X = Mi3 )
        | ( X = Ma )
        | $let(
            pos: nat,
            pos:= 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)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_7949_vebt__member_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya)),X)
    <=> $ite(
          X = Mi3,
          $true,
          $ite(
            X = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Mi3),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),X),
                $false,
                $let(
                  h: nat,
                  h:= 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)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_7950_XOR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(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))),Na)) ) ) ) ).

% XOR_upper
tff(fact_7951_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_7952_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = 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(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_7953_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(uminus_uminus_rat(A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_7954_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7955_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7956_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7957_gbinomial__pochhammer,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(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,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_7958_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(A)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_7959_gbinomial__pochhammer,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_7960_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_7961_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_7962_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_7963_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa = zero_zero_nat,
                    (A4),
                    $ite(Xa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_7964_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xa = zero_zero_nat,
                  (A4),
                  $ite(Xa = one_one_nat,(B4),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3)
             => ~ $let(
                    pos: nat,
                    pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_7965_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xa = zero_zero_nat,
                (A4),
                $ite(Xa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3)
               => $let(
                    pos: nat,
                    pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_7966_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( ! [Mi: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa = Mi )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xa = Mi )
                  | ( Xa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
               => ~ $let(
                      pos: nat,
                      pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_7967_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_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_7968_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(X),Xa)
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xa = zero_zero_nat,
                  (A4),
                  $ite(Xa = one_one_nat,(B4),$false) ) )
       => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
             => ~ $ite(
                    Xa = Mi,
                    $true,
                    $ite(
                      Xa = Ma2,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xa),
                          $false,
                          $let(
                            h: nat,
                            h:= 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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_7969_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xa = Mi )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xa = Mi )
                        | ( Xa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_7970_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( ! [Uu2: $o,Uv2: $o] : X != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa = Mi )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xa = Mi )
                    | ( Xa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => $let(
                        pos: nat,
                        pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_7971_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(groups3417619833198082522nteger(zero_n356916108424825756nteger,numera6620942414471956472nteger(aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7972_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,groups9119017779487936845_o_nat(zero_n2687167440665602831ol_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7973_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),Na) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7974_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(X),Xa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( X = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xa = zero_zero_nat,
                    (A4),
                    $ite(Xa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xa = Mi,
                            $true,
                            $ite(
                              Xa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= 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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_7975_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_7976_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            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(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_7977_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_7978_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(
        K = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_7979_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(
        K = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_7980_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => topolo6980174941875973593q_real(aTP_Lamm_bj(X)) ) ).

% monoseq_arctan_series
tff(fact_7981_ln__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aTP_Lamm_bk(X)) ) ) ) ).

% ln_series
tff(fact_7982_arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(arctan,X) = suminf_real(aTP_Lamm_bl(X)) ) ) ).

% arctan_series
tff(fact_7983_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => summable_real(aTP_Lamm_bl(X)) ) ).

% summable_arctan_series
tff(fact_7984_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_bm(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7985_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_bn(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7986_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),X))
     => ( aa_real_o(aa_real_fun_real_o(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_bp(F),Z)) ) ) ).

% powser_insidea
tff(fact_7987_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_br(F),Z)) ) ) ).

% powser_insidea
tff(fact_7988_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),X))
     => ( aa_real_o(aa_real_fun_real_o(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_bo(F),Z)) ) ) ).

% powser_inside
tff(fact_7989_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z)) ) ) ).

% powser_inside
tff(fact_7990_summable__exp,axiom,
    ! [X: complex] : summable_complex(aTP_Lamm_bs(X)) ).

% summable_exp
tff(fact_7991_summable__exp,axiom,
    ! [X: real] : summable_real(aTP_Lamm_bt(X)) ).

% summable_exp
tff(fact_7992_monoseq__realpow,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => topolo6980174941875973593q_real(power_power_real(X)) ) ) ).

% monoseq_realpow
tff(fact_7993_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_bu) ).

% pi_series
tff(fact_7994_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(power_power_real(C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7995_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7996_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bv(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7997_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_bw(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7998_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_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_7999_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(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_8000_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bx(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_8001_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_by(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8002_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bz(I),F)) ).

% summable_single
tff(fact_8003_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ca(I),F)) ).

% summable_single
tff(fact_8004_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cb(I),F)) ).

% summable_single
tff(fact_8005_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_cc(I),F)) ).

% summable_single
tff(fact_8006_summable__zero,axiom,
    summable_complex(aTP_Lamm_cd) ).

% summable_zero
tff(fact_8007_summable__zero,axiom,
    summable_real(aTP_Lamm_ce) ).

% summable_zero
tff(fact_8008_summable__zero,axiom,
    summable_nat(aTP_Lamm_ar) ).

% summable_zero
tff(fact_8009_summable__zero,axiom,
    summable_int(aTP_Lamm_cf) ).

% summable_zero
tff(fact_8010_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_cd) = zero_zero_complex ).

% suminf_zero
tff(fact_8011_suminf__zero,axiom,
    suminf_real(aTP_Lamm_ce) = zero_zero_real ).

% suminf_zero
tff(fact_8012_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_ar) = zero_zero_nat ).

% suminf_zero
tff(fact_8013_suminf__zero,axiom,
    suminf_int(aTP_Lamm_cf) = zero_zero_int ).

% suminf_zero
tff(fact_8014_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_8015_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_ch(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_8016_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aTP_Lamm_ci(C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8017_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(F),G)) ) ) ).

% summable_add
tff(fact_8018_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ck(F),G)) ) ) ).

% summable_add
tff(fact_8019_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_cl(F),G)) ) ) ).

% summable_add
tff(fact_8020_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bv(F),C)) ) ).

% summable_divide
tff(fact_8021_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_bw(F),C)) ) ).

% summable_divide
tff(fact_8022_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_cm(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_8023_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_8024_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_8025_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( summable_int(F)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_8026_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( summable_real(F)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_8027_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bx(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_8028_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_by(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8029_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_8030_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_8031_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_8032_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(F),G)) ) ) ) ).

% suminf_add
tff(fact_8033_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ck(F),G)) ) ) ) ).

% suminf_add
tff(fact_8034_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_cl(F),G)) ) ) ) ).

% suminf_add
tff(fact_8035_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_bv(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_8036_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_bw(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_8037_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N4: nat] : aa_nat_nat(F,N4) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8038_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N4: nat] : aa_nat_int(F,N4) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8039_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N4: nat] : aa_nat_real(F,N4) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_8040_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_nonneg
tff(fact_8041_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_nonneg
tff(fact_8042_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_nonneg
tff(fact_8043_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_pos
tff(fact_8044_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_pos
tff(fact_8045_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_pos
tff(fact_8046_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_bm(F)) ).

% summable_zero_power'
tff(fact_8047_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_bn(F)) ).

% summable_zero_power'
tff(fact_8048_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_cn(F)) ).

% summable_zero_power'
tff(fact_8049_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_bm(F)) ).

% summable_0_powser
tff(fact_8050_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_bn(F)) ).

% summable_0_powser
tff(fact_8051_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_co(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8052_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_cp(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8053_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_co(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8054_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cp(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8055_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_cq(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8056_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_cr(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8057_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_cs(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8058_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ct(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8059_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_8060_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_8061_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_8062_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_8063_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_8064_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_8065_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_8066_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_8067_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_8068_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_8069_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real)
     => summable_real(power_power_real(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8070_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real)
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_8071_summable__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => summable_real(power_power_real(C)) ) ).

% summable_geometric
tff(fact_8072_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_8073_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_cm(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_8074_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(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_co(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8075_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_bo(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_cp(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8076_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_co(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8077_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_cp(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8078_suminf__exist__split,axiom,
    ! [R4: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
     => ( summable_real(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),N7)))),R4) ) ) ) ).

% suminf_exist_split
tff(fact_8079_suminf__exist__split,axiom,
    ! [R4: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
     => ( summable_complex(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_cu(F),N7)))),R4) ) ) ) ).

% suminf_exist_split
tff(fact_8080_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8081_Abel__lemma,axiom,
    ! [R4: real,R0: real,A: fun_nat_complex,M7: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R4)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R4),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(power_power_real(R0),N))),M7)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_cv(R4),A)) ) ) ) ).

% Abel_lemma
tff(fact_8082_summable__ratio__test,axiom,
    ! [C: real,N2: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8083_summable__ratio__test,axiom,
    ! [C: real,N2: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8084_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_8085_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      $let(
                        half: nat,
                        half:= aa_nat_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(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))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_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(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,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_8086_sin__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_cw(X)),aa_real_real(sin_real,X)) ).

% sin_paired
tff(fact_8087_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xa))
               => $ite(
                    Xa = zero_zero_nat,
                    (A4),
                    $ite(Xa = one_one_nat,(B4),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa)) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xa))
                       => $ite(
                            Xa = Mi,
                            $true,
                            $ite(
                              Xa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= 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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_8088_intind,axiom,
    ! [I: nat,Na: nat,P: fun_nat_o,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_o(P,X)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Na,X)),I)) ) ) ).

% intind
tff(fact_8089_intind,axiom,
    ! [I: nat,Na: nat,P: fun_int_o,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_int_o(P,X)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Na,X)),I)) ) ) ).

% intind
tff(fact_8090_intind,axiom,
    ! [I: nat,Na: nat,P: fun_VEBT_VEBT_o,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_VEBT_VEBT_o(P,X)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Na,X),I)) ) ) ).

% intind
tff(fact_8091_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_8092_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_8093_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X) = replicate_VEBT_VEBT(Na,Y) )
    <=> ( ( M = Na )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_8094_length__replicate,axiom,
    ! [Na: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Na,X)) = Na ).

% length_replicate
tff(fact_8095_length__replicate,axiom,
    ! [Na: nat,X: $o] : size_size_list_o(replicate_o(Na,(X))) = Na ).

% length_replicate
tff(fact_8096_length__replicate,axiom,
    ! [Na: nat,X: nat] : size_size_list_nat(replicate_nat(Na,X)) = Na ).

% length_replicate
tff(fact_8097_length__replicate,axiom,
    ! [Na: nat,X: int] : size_size_list_int(replicate_int(Na,X)) = Na ).

% length_replicate
tff(fact_8098_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_8099_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_8100_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_cd),zero_zero_complex) ).

% sums_zero
tff(fact_8101_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_ce),zero_zero_real) ).

% sums_zero
tff(fact_8102_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_ar),zero_zero_nat) ).

% sums_zero
tff(fact_8103_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_cf),zero_zero_int) ).

% sums_zero
tff(fact_8104_bit_Oconj__cancel__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = zero_zero_int ).

% bit.conj_cancel_left
tff(fact_8105_bit_Oconj__cancel__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = zero_zero_int ).

% bit.conj_cancel_right
tff(fact_8106_Ball__set__replicate,axiom,
    ! [Na: nat,A: int,P: fun_int_o] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(replicate_int(Na,A)))
         => aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8107_Ball__set__replicate,axiom,
    ! [Na: nat,A: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Na,A)))
         => aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8108_Ball__set__replicate,axiom,
    ! [Na: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,A)))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8109_Bex__set__replicate,axiom,
    ! [Na: nat,A: int,P: fun_int_o] :
      ( ? [X4: int] :
          ( member_int(X4,set_int2(replicate_int(Na,A)))
          & aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8110_Bex__set__replicate,axiom,
    ! [Na: nat,A: nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Na,A)))
          & aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8111_Bex__set__replicate,axiom,
    ! [Na: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,A)))
          & aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8112_in__set__replicate,axiom,
    ! [X: complex,Na: nat,Y: complex] :
      ( member_complex(X,set_complex2(replicate_complex(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8113_in__set__replicate,axiom,
    ! [X: real,Na: nat,Y: real] :
      ( member_real(X,set_real2(replicate_real(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8114_in__set__replicate,axiom,
    ! [X: set_nat,Na: nat,Y: set_nat] :
      ( member_set_nat(X,set_set_nat2(replicate_set_nat(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8115_in__set__replicate,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( member_int(X,set_int2(replicate_int(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8116_in__set__replicate,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8117_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,Y)))
    <=> ( ( X = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8118_nth__replicate,axiom,
    ! [I: nat,Na: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_nat(nth_nat(replicate_nat(Na,X)),I) = X ) ) ).

% nth_replicate
tff(fact_8119_nth__replicate,axiom,
    ! [I: nat,Na: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_int(nth_int(replicate_int(Na,X)),I) = X ) ) ).

% nth_replicate
tff(fact_8120_nth__replicate,axiom,
    ! [I: nat,Na: nat,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Na,X),I) = X ) ) ).

% nth_replicate
tff(fact_8121_bit_Ocompl__one,axiom,
    bit_ri7632146776885996613nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_8122_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_8123_bit_Ocompl__zero,axiom,
    bit_ri7632146776885996613nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_8124_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_8125_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_8126_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_8127_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_8128_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_8129_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_8130_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_8131_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_8132_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_8133_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_8134_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_8135_minus__not__numeral__eq,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_ri7632146776885996613nteger(numera6620942414471956472nteger(Na))) = numera6620942414471956472nteger(inc(Na)) ).

% minus_not_numeral_eq
tff(fact_8136_minus__not__numeral__eq,axiom,
    ! [Na: num] : aa_int_int(uminus_uminus_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,inc(Na)) ).

% minus_not_numeral_eq
tff(fact_8137_even__not__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri7632146776885996613nteger(A))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_not_iff
tff(fact_8138_even__not__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,A))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_not_iff
tff(fact_8139_push__bit__minus__one__eq__not__mask,axiom,
    ! [Na: nat] : bit_se7788150548672797655nteger(Na,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(aa_nat_Code_integer(bit_se2119862282449309892nteger,Na)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_8140_push__bit__minus__one__eq__not__mask,axiom,
    ! [Na: nat] : aa_int_int(bit_se545348938243370406it_int(Na),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_8141_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_bm(A)),X)
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8142_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( aa_real_o(sums_real(aTP_Lamm_bn(A)),X)
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_8143_not__one__eq,axiom,
    bit_ri7632146776885996613nteger(one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_8144_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_8145_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,Tb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( aa_nat_o(sums_nat(F),S)
       => ( aa_nat_o(sums_nat(G),Tb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),Tb) ) ) ) ).

% sums_le
tff(fact_8146_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,Tb: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( aa_int_o(sums_int(F),S)
       => ( aa_int_o(sums_int(G),Tb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S),Tb) ) ) ) ).

% sums_le
tff(fact_8147_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,Tb: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( aa_real_o(sums_real(F),S)
       => ( aa_real_o(sums_real(G),Tb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S),Tb) ) ) ) ).

% sums_le
tff(fact_8148_sums__0,axiom,
    ! [F: fun_nat_complex] :
      ( ! [N: nat] : aa_nat_complex(F,N) = zero_zero_complex
     => aa_complex_o(sums_complex(F),zero_zero_complex) ) ).

% sums_0
tff(fact_8149_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N: nat] : aa_nat_real(F,N) = zero_zero_real
     => aa_real_o(sums_real(F),zero_zero_real) ) ).

% sums_0
tff(fact_8150_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_nat(F,N) = zero_zero_nat
     => aa_nat_o(sums_nat(F),zero_zero_nat) ) ).

% sums_0
tff(fact_8151_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N: nat] : aa_nat_int(F,N) = zero_zero_int
     => aa_int_o(sums_int(F),zero_zero_int) ) ).

% sums_0
tff(fact_8152_sums__single,axiom,
    ! [I: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bz(I),F)),aa_nat_complex(F,I)) ).

% sums_single
tff(fact_8153_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ca(I),F)),aa_nat_real(F,I)) ).

% sums_single
tff(fact_8154_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cb(I),F)),aa_nat_nat(F,I)) ).

% sums_single
tff(fact_8155_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_cc(I),F)),aa_nat_int(F,I)) ).

% sums_single
tff(fact_8156_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% sums_add
tff(fact_8157_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ck(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_8158_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_cl(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_8159_sums__divide,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_bv(F),C)),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ).

% sums_divide
tff(fact_8160_sums__divide,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_bw(F),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ).

% sums_divide
tff(fact_8161_bit__not__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),Na)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Na) ) ).

% bit_not_int_iff
tff(fact_8162_take__bit__not__iff,axiom,
    ! [Na: nat,A: int,B: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_ri7919022796975470100ot_int,B)) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Na),A) = aa_int_int(bit_se2923211474154528505it_int(Na),B) ) ) ).

% take_bit_not_iff
tff(fact_8163_take__bit__not__take__bit,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se2923211474154528505it_int(Na),A))) = aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_ri7919022796975470100ot_int,A)) ).

% take_bit_not_take_bit
tff(fact_8164_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_8165_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_8166_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_8167_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_8168_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bx(C),F)),aa_complex_complex(times_times_complex(C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_8169_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_by(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_8170_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_cx(C),F)),aa_complex_complex(times_times_complex(D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8171_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cy(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8172_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => ( X2 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8173_replicate__length__same,axiom,
    ! [Xs: list_o,X: $o] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => ( (X2)
          <=> (X) ) )
     => ( replicate_o(size_size_list_o(Xs),(X)) = Xs ) ) ).

% replicate_length_same
tff(fact_8174_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( X2 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8175_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X2: int] :
          ( member_int(X2,set_int2(Xs))
         => ( X2 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_8176_replicate__eqI,axiom,
    ! [Xs: list_complex,Na: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Na )
     => ( ! [Y2: complex] :
            ( member_complex(Y2,set_complex2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_complex(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_8177_replicate__eqI,axiom,
    ! [Xs: list_real,Na: nat,X: real] :
      ( ( size_size_list_real(Xs) = Na )
     => ( ! [Y2: real] :
            ( member_real(Y2,set_real2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_real(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_8178_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Na: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Na )
     => ( ! [Y2: set_nat] :
            ( member_set_nat(Y2,set_set_nat2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_set_nat(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_8179_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Na: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Na )
     => ( ! [Y2: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y2,set_VEBT_VEBT2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_VEBT_VEBT(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_8180_replicate__eqI,axiom,
    ! [Xs: list_o,Na: nat,X: $o] :
      ( ( size_size_list_o(Xs) = Na )
     => ( ! [Y2: $o] :
            ( member_o((Y2),set_o2(Xs))
           => ( (Y2)
            <=> (X) ) )
       => ( Xs = replicate_o(Na,(X)) ) ) ) ).

% replicate_eqI
tff(fact_8181_replicate__eqI,axiom,
    ! [Xs: list_nat,Na: nat,X: nat] :
      ( ( size_size_list_nat(Xs) = Na )
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_nat(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_8182_replicate__eqI,axiom,
    ! [Xs: list_int,Na: nat,X: int] :
      ( ( size_size_list_int(Xs) = Na )
     => ( ! [Y2: int] :
            ( member_int(Y2,set_int2(Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate_int(Na,X) ) ) ) ).

% replicate_eqI
tff(fact_8183_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bx(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).

% sums_mult_D
tff(fact_8184_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_by(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% sums_mult_D
tff(fact_8185_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_cz(F)),S)
       => aa_complex_o(sums_complex(F),S) ) ) ).

% sums_Suc_imp
tff(fact_8186_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_cm(F)),S)
       => aa_real_o(sums_real(F),S) ) ) ).

% sums_Suc_imp
tff(fact_8187_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( aa_real_o(sums_real(aTP_Lamm_cm(F)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8188_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_cm(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8189_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_da(F)),L)
     => aa_nat_o(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8190_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_db(F)),L)
     => aa_int_o(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8191_sums__zero__iff__shift,axiom,
    ! [Na: nat,F: fun_nat_complex,S: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ( aa_nat_complex(F,I2) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_dc(Na),F)),S)
      <=> aa_complex_o(sums_complex(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_8192_sums__zero__iff__shift,axiom,
    ! [Na: nat,F: fun_nat_real,S: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_dd(Na),F)),S)
      <=> aa_real_o(sums_real(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_8193_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_8194_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_8195_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_8196_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_8197_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_8198_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_8199_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_8200_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_8201_disjunctive__diff,axiom,
    ! [B: int,A: int] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(B),N)
         => aa_nat_o(bit_se1146084159140164899it_int(A),N) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_ri7919022796975470100ot_int,B)) ) ) ).

% disjunctive_diff
tff(fact_8202_take__bit__not__eq__mask__diff,axiom,
    ! [Na: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Na),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),aa_int_int(bit_se2923211474154528505it_int(Na),A)) ).

% take_bit_not_eq_mask_diff
tff(fact_8203_minus__numeral__inc__eq,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(Na))) = bit_ri7632146776885996613nteger(numera6620942414471956472nteger(Na)) ).

% minus_numeral_inc_eq
tff(fact_8204_minus__numeral__inc__eq,axiom,
    ! [Na: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(Na))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Na)) ).

% minus_numeral_inc_eq
tff(fact_8205_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_de(M),Z)),aa_nat_complex(power_power_complex(Z),M)) ).

% powser_sums_if
tff(fact_8206_powser__sums__if,axiom,
    ! [M: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_df(M),Z)),aa_nat_real(power_power_real(Z),M)) ).

% powser_sums_if
tff(fact_8207_powser__sums__if,axiom,
    ! [M: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_dg(M),Z)),aa_nat_int(power_power_int(Z),M)) ).

% powser_sums_if
tff(fact_8208_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_bm(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8209_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_bn(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8210_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_8211_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_not_iff_int
tff(fact_8212_and__not__numerals_I2_J,axiom,
    ! [Na: 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,Na)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_8213_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_8214_not__numeral__Bit0__eq,axiom,
    ! [Na: num] : bit_ri7632146776885996613nteger(numera6620942414471956472nteger(aa_num_num(bit0,Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,Na))) ).

% not_numeral_Bit0_eq
tff(fact_8215_not__numeral__Bit0__eq,axiom,
    ! [Na: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))) ).

% not_numeral_Bit0_eq
tff(fact_8216_bit__minus__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),Na)
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),Na) ) ).

% bit_minus_int_iff
tff(fact_8217_not__numeral__BitM__eq,axiom,
    ! [Na: num] : bit_ri7632146776885996613nteger(numera6620942414471956472nteger(bitM(Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,Na))) ).

% not_numeral_BitM_eq
tff(fact_8218_not__numeral__BitM__eq,axiom,
    ! [Na: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bitM(Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) ).

% not_numeral_BitM_eq
tff(fact_8219_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Na))) = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
tff(fact_8220_push__bit__mask__eq,axiom,
    ! [M: nat,Na: nat] : aa_int_int(bit_se545348938243370406it_int(M),aa_nat_int(bit_se2000444600071755411sk_int,Na)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,M))) ).

% push_bit_mask_eq
tff(fact_8221_unset__bit__eq__and__not,axiom,
    ! [Na: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),A) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Na),one_one_int))) ).

% unset_bit_eq_and_not
tff(fact_8222_unset__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K) = aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Na),one_one_int))) ).

% unset_bit_int_def
tff(fact_8223_and__not__numerals_I5_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na)))) ).

% and_not_numerals(5)
tff(fact_8224_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_8225_and__not__numerals_I3_J,axiom,
    ! [Na: 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,Na)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_8226_and__not__numerals_I6_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na)))) ).

% and_not_numerals(6)
tff(fact_8227_and__not__numerals_I9_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na)))) ).

% and_not_numerals(9)
tff(fact_8228_bit__not__iff__eq,axiom,
    ! [A: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,A)),Na)
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int )
        & ~ aa_nat_o(bit_se1146084159140164899it_int(A),Na) ) ) ).

% bit_not_iff_eq
tff(fact_8229_minus__exp__eq__not__mask,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Na)) = bit_ri7632146776885996613nteger(aa_nat_Code_integer(bit_se2119862282449309892nteger,Na)) ).

% minus_exp_eq_not_mask
tff(fact_8230_minus__exp__eq__not__mask,axiom,
    ! [Na: nat] : aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% minus_exp_eq_not_mask
tff(fact_8231_geometric__sums,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => aa_real_o(sums_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))) ) ).

% geometric_sums
tff(fact_8232_geometric__sums,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => aa_complex_o(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C))) ) ).

% geometric_sums
tff(fact_8233_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_dh),one_one_real) ).

% power_half_series
tff(fact_8234_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( aa_real_o(sums_real(G),X)
     => aa_real_o(sums_real(aTP_Lamm_di(G)),X) ) ).

% sums_if'
tff(fact_8235_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),X)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_dj(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ) ) ).

% sums_if
tff(fact_8236_and__not__numerals_I8_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na))))) ).

% and_not_numerals(8)
tff(fact_8237_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_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(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(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_8238_cos__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_dk(X)),aa_real_real(cos_real,X)) ).

% cos_paired
tff(fact_8239_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
        $let(
          half: nat,
          half:= aa_nat_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(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))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_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(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,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_8240_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xa))
               => ~ $ite(
                      Xa = zero_zero_nat,
                      (A4),
                      $ite(Xa = one_one_nat,(B4),$false) ) ) )
         => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xa))
                 => ~ $ite(
                        Xa = Mi,
                        $true,
                        $ite(
                          Xa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xa),
                              $false,
                              $let(
                                h: nat,
                                h:= 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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_8241_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aTP_Lamm_dl(Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8242_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => aa_complex_o(sums_complex(aTP_Lamm_dm(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8243_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(X),Xa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xa = zero_zero_nat,
                      (A4),
                      $ite(Xa = one_one_nat,(B4),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xa = Mi,
                              $true,
                              $ite(
                                Xa = Ma2,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= 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)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_8244_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xa = zero_zero_nat,
                      (A4),
                      $ite(Xa = one_one_nat,(B4),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),Xa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_8245_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xa))
               => ~ $ite(
                      Xa = zero_zero_nat,
                      (A4),
                      $ite(Xa = one_one_nat,(B4),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),Xa))
                 => ~ $let(
                        pos: nat,
                        pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_8246_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [A4: $o,B4: $o] :
              ( ( X = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xa))
               => $ite(
                    Xa = zero_zero_nat,
                    (A4),
                    $ite(Xa = one_one_nat,(B4),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S3),Xa))
                   => $let(
                        pos: nat,
                        pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_8247_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),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)) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2),Xa))
                   => ( ( Xa = Mi )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa))
                     => ( ( Xa = Mi )
                        | ( Xa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xa))
                       => $let(
                            pos: nat,
                            pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_8248_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2),Xa))
               => ~ ( ( Xa = Mi )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa))
                 => ~ ( ( Xa = Mi )
                      | ( Xa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xa))
                   => ~ $let(
                          pos: nat,
                          pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_8249_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa)) ) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xa = Mi )
                        | ( Xa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3,Vb2),Xa)) ) )
             => ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xa = Mi )
                          | ( Xa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= 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,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_8250_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_dn(C),X))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_do(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_dn(C),X))) ) ).

% diffs_equiv
tff(fact_8251_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_dp(C),X))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_dq(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_dp(C),X))) ) ).

% diffs_equiv
tff(fact_8252_exp__first__two__terms,axiom,
    ! [X: 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)),suminf_real(aTP_Lamm_dr(X))) ).

% exp_first_two_terms
tff(fact_8253_exp__first__two__terms,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),X)),suminf_complex(aTP_Lamm_ds(X))) ).

% exp_first_two_terms
tff(fact_8254_monoI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M4)),aa_nat_set_nat(X6,N)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI1
tff(fact_8255_monoI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M4)),aa_nat_rat(X6,N)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI1
tff(fact_8256_monoI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_num_o(ord_less_eq_num(aa_nat_num(X6,M4)),aa_nat_num(X6,N)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI1
tff(fact_8257_monoI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M4)),aa_nat_nat(X6,N)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI1
tff(fact_8258_monoI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M4)),aa_nat_int(X6,N)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI1
tff(fact_8259_monoI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M4)),aa_nat_real(X6,N)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI1
tff(fact_8260_monoI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,M4)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI2
tff(fact_8261_monoI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,M4)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI2
tff(fact_8262_monoI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_num_o(ord_less_eq_num(aa_nat_num(X6,N)),aa_nat_num(X6,M4)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI2
tff(fact_8263_monoI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,M4)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI2
tff(fact_8264_monoI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,M4)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI2
tff(fact_8265_monoI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,M4)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI2
tff(fact_8266_scaleR__cancel__right,axiom,
    ! [A: real,X: complex,B: real] :
      ( ( aa_complex_complex(real_V2046097035970521341omplex(A),X) = aa_complex_complex(real_V2046097035970521341omplex(B),X) )
    <=> ( ( A = B )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_cancel_right
tff(fact_8267_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X) )
    <=> ( ( A = B )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
tff(fact_8268_scaleR__zero__right,axiom,
    ! [A: real] : aa_complex_complex(real_V2046097035970521341omplex(A),zero_zero_complex) = zero_zero_complex ).

% scaleR_zero_right
tff(fact_8269_scaleR__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),zero_zero_real) = zero_zero_real ).

% scaleR_zero_right
tff(fact_8270_scaleR__zero__left,axiom,
    ! [X: complex] : aa_complex_complex(real_V2046097035970521341omplex(zero_zero_real),X) = zero_zero_complex ).

% scaleR_zero_left
tff(fact_8271_scaleR__zero__left,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,zero_zero_real),X) = zero_zero_real ).

% scaleR_zero_left
tff(fact_8272_scaleR__eq__0__iff,axiom,
    ! [A: real,X: complex] :
      ( ( aa_complex_complex(real_V2046097035970521341omplex(A),X) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_eq_0_iff
tff(fact_8273_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
tff(fact_8274_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),B)) )
    <=> ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_8275_scaleR__power,axiom,
    ! [X: real,Y: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,X),Y)),Na) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(power_power_real(X),Na)),aa_nat_real(power_power_real(Y),Na)) ).

% scaleR_power
tff(fact_8276_scaleR__power,axiom,
    ! [X: real,Y: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(real_V2046097035970521341omplex(X),Y)),Na) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(power_power_real(X),Na)),aa_nat_complex(power_power_complex(Y),Na)) ).

% scaleR_power
tff(fact_8277_scaleR__minus1__left,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,one_one_real)),X) = aa_real_real(uminus_uminus_real,X) ).

% scaleR_minus1_left
tff(fact_8278_scaleR__minus1__left,axiom,
    ! [X: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(uminus_uminus_real,one_one_real)),X) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% scaleR_minus1_left
tff(fact_8279_scaleR__collapse,axiom,
    ! [U: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),U)),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A)) = A ).

% scaleR_collapse
tff(fact_8280_scaleR__times,axiom,
    ! [U: num,W: num,A: complex] : aa_complex_complex(real_V2046097035970521341omplex(numeral_numeral_real(U)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),A) ).

% scaleR_times
tff(fact_8281_scaleR__times,axiom,
    ! [U: num,W: num,A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,numeral_numeral_real(U)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),A) ).

% scaleR_times
tff(fact_8282_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(W)),numeral_numeral_real(V))),A) ).

% inverse_scaleR_times
tff(fact_8283_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(W)),numeral_numeral_real(V))),A) ).

% inverse_scaleR_times
tff(fact_8284_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(U)),numeral_numeral_real(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),numeral_numeral_real(V))),A) ).

% fraction_scaleR_times
tff(fact_8285_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(U)),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),numeral_numeral_real(V))),A) ).

% fraction_scaleR_times
tff(fact_8286_scaleR__half__double,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)) = A ).

% scaleR_half_double
tff(fact_8287_scaleR__right__imp__eq,axiom,
    ! [X: complex,A: real,B: real] :
      ( ( X != zero_zero_complex )
     => ( ( aa_complex_complex(real_V2046097035970521341omplex(A),X) = aa_complex_complex(real_V2046097035970521341omplex(B),X) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_8288_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_8289_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ).

% scaleR_right_distrib
tff(fact_8290_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,X),Xa)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Y),Xa)) ).

% scaleR_left.add
tff(fact_8291_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X)) ).

% scaleR_left_distrib
tff(fact_8292_of__real__def,axiom,
    ! [R4: real] : aa_real_real(real_V1803761363581548252l_real,R4) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,R4),one_one_real) ).

% of_real_def
tff(fact_8293_of__real__def,axiom,
    ! [R4: real] : aa_real_complex(real_V4546457046886955230omplex,R4) = aa_complex_complex(real_V2046097035970521341omplex(R4),one_one_complex) ).

% of_real_def
tff(fact_8294_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),C)) ) ) ).

% scaleR_right_mono_neg
tff(fact_8295_scaleR__right__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X)) ) ) ).

% scaleR_right_mono
tff(fact_8296_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_8297_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_8298_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_8299_scaleR__left__mono,axiom,
    ! [X: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ) ) ).

% scaleR_left_mono
tff(fact_8300_scaleR__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B)) ) ) ).

% scaleR_left_mono_neg
tff(fact_8301_eq__vector__fraction__iff,axiom,
    ! [X: complex,U: real,V: real,A: complex] :
      ( ( X = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) )
    <=> $ite(V = zero_zero_real,X = zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(V),X) = aa_complex_complex(real_V2046097035970521341omplex(U),A)) ) ).

% eq_vector_fraction_iff
tff(fact_8302_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) )
    <=> $ite(V = zero_zero_real,X = zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,V),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A)) ) ).

% eq_vector_fraction_iff
tff(fact_8303_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,X: complex] :
      ( ( aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) = X )
    <=> $ite(V = zero_zero_real,X = zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(U),A) = aa_complex_complex(real_V2046097035970521341omplex(V),X)) ) ).

% vector_fraction_eq_iff
tff(fact_8304_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) = X )
    <=> $ite(V = zero_zero_real,X = zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,V),X)) ) ).

% vector_fraction_eq_iff
tff(fact_8305_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_8306_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_8307_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_8308_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_8309_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_8310_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_8311_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_8312_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_8313_split__scaleR__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)) ) ).

% split_scaleR_pos_le
tff(fact_8314_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real) ) ).

% split_scaleR_neg_le
tff(fact_8315_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),D)) ) ) ) ) ).

% scaleR_mono'
tff(fact_8316_scaleR__mono,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Y)) ) ) ) ) ).

% scaleR_mono
tff(fact_8317_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),X) ) ) ).

% scaleR_left_le_one_le
tff(fact_8318_scaleR__2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,numeral_numeral_real(aa_num_num(bit0,one))),X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% scaleR_2
tff(fact_8319_real__vector__affinity__eq,axiom,
    ! [M: real,X: real,C: real,Y: real] :
      ( ( M != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,M),X)),C) = Y )
      <=> ( X = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,M)),Y)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,M)),C)) ) ) ) ).

% real_vector_affinity_eq
tff(fact_8320_real__vector__eq__affinity,axiom,
    ! [M: real,Y: real,X: real,C: real] :
      ( ( M != zero_zero_real )
     => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,M),X)),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,M)),Y)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,M)),C)) = X ) ) ) ).

% real_vector_eq_affinity
tff(fact_8321_neg__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_le_divideR_eq
tff(fact_8322_neg__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% neg_divideR_le_eq
tff(fact_8323_pos__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% pos_le_divideR_eq
tff(fact_8324_pos__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_divideR_le_eq
tff(fact_8325_pos__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_divideR_less_eq
tff(fact_8326_pos__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% pos_less_divideR_eq
tff(fact_8327_neg__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% neg_divideR_less_eq
tff(fact_8328_neg__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_less_divideR_eq
tff(fact_8329_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,X)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_8330_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: complex] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(real_V2046097035970521341omplex(A),X)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,A)),aa_complex_complex(invers8013647133539491842omplex,X)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_8331_diffs__def,axiom,
    ! [C: fun_nat_real,X3: nat] : aa_nat_real(diffs_real(C),X3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X3))),aa_nat_real(C,aa_nat_nat(suc,X3))) ).

% diffs_def
tff(fact_8332_diffs__def,axiom,
    ! [C: fun_nat_int,X3: nat] : aa_nat_int(diffs_int(C),X3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X3))),aa_nat_int(C,aa_nat_nat(suc,X3))) ).

% diffs_def
tff(fact_8333_diffs__def,axiom,
    ! [C: fun_nat_complex,X3: nat] : aa_nat_complex(diffs_complex(C),X3) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X3))),aa_nat_complex(C,aa_nat_nat(suc,X3))) ).

% diffs_def
tff(fact_8334_diffs__def,axiom,
    ! [C: fun_nat_rat,X3: nat] : aa_nat_rat(diffs_rat(C),X3) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X3))),aa_nat_rat(C,aa_nat_nat(suc,X3))) ).

% diffs_def
tff(fact_8335_summable__exp__generic,axiom,
    ! [X: real] : summable_real(aTP_Lamm_dt(X)) ).

% summable_exp_generic
tff(fact_8336_summable__exp__generic,axiom,
    ! [X: complex] : summable_complex(aTP_Lamm_du(X)) ).

% summable_exp_generic
tff(fact_8337_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( ! [X2: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),X2))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_dp(C),X)) ) ).

% termdiff_converges_all
tff(fact_8338_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,X: real] :
      ( ! [X2: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),X2))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_dn(C),X)) ) ).

% termdiff_converges_all
tff(fact_8339_sin__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_dv(X)),aa_real_real(sin_real,X)) ).

% sin_converges
tff(fact_8340_sin__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_dw(X)),aa_complex_complex(sin_complex,X)) ).

% sin_converges
tff(fact_8341_sin__def,axiom,
    ! [X3: real] : aa_real_real(sin_real,X3) = suminf_real(aTP_Lamm_dv(X3)) ).

% sin_def
tff(fact_8342_sin__def,axiom,
    ! [X3: complex] : aa_complex_complex(sin_complex,X3) = suminf_complex(aTP_Lamm_dw(X3)) ).

% sin_def
tff(fact_8343_cos__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_dx(X)),aa_real_real(cos_real,X)) ).

% cos_converges
tff(fact_8344_cos__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_dy(X)),aa_complex_complex(cos_complex,X)) ).

% cos_converges
tff(fact_8345_cos__def,axiom,
    ! [X3: real] : aa_real_real(cos_real,X3) = suminf_real(aTP_Lamm_dx(X3)) ).

% cos_def
tff(fact_8346_cos__def,axiom,
    ! [X3: complex] : aa_complex_complex(cos_complex,X3) = suminf_complex(aTP_Lamm_dy(X3)) ).

% cos_def
tff(fact_8347_summable__norm__sin,axiom,
    ! [X: real] : summable_real(aTP_Lamm_dz(X)) ).

% summable_norm_sin
tff(fact_8348_summable__norm__sin,axiom,
    ! [X: complex] : summable_real(aTP_Lamm_ea(X)) ).

% summable_norm_sin
tff(fact_8349_summable__norm__cos,axiom,
    ! [X: real] : summable_real(aTP_Lamm_eb(X)) ).

% summable_norm_cos
tff(fact_8350_summable__norm__cos,axiom,
    ! [X: complex] : summable_real(aTP_Lamm_ec(X)) ).

% summable_norm_cos
tff(fact_8351_pos__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_8352_pos__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_8353_neg__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_8354_neg__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_8355_neg__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_8356_neg__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_8357_pos__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_8358_pos__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_8359_exp__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_dt(X)),aa_real_real(exp_real,X)) ).

% exp_converges
tff(fact_8360_exp__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_du(X)),aa_complex_complex(exp_complex,X)) ).

% exp_converges
tff(fact_8361_exp__def,axiom,
    ! [X3: real] : aa_real_real(exp_real,X3) = suminf_real(aTP_Lamm_dt(X3)) ).

% exp_def
tff(fact_8362_exp__def,axiom,
    ! [X3: complex] : aa_complex_complex(exp_complex,X3) = suminf_complex(aTP_Lamm_du(X3)) ).

% exp_def
tff(fact_8363_summable__norm__exp,axiom,
    ! [X: real] : summable_real(aTP_Lamm_ed(X)) ).

% summable_norm_exp
tff(fact_8364_summable__norm__exp,axiom,
    ! [X: complex] : summable_real(aTP_Lamm_ee(X)) ).

% summable_norm_exp
tff(fact_8365_sin__minus__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_ef(X)),aa_real_real(sin_real,X)) ).

% sin_minus_converges
tff(fact_8366_sin__minus__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_eg(X)),aa_complex_complex(sin_complex,X)) ).

% sin_minus_converges
tff(fact_8367_cos__minus__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_eh(X)),aa_real_real(cos_real,X)) ).

% cos_minus_converges
tff(fact_8368_cos__minus__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_ei(X)),aa_complex_complex(cos_complex,X)) ).

% cos_minus_converges
tff(fact_8369_termdiff__converges,axiom,
    ! [X: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K5)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),X2)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(X),C)) ) ) ).

% termdiff_converges
tff(fact_8370_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5)
     => ( ! [X2: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),X2)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ek(X),C)) ) ) ).

% termdiff_converges
tff(fact_8371_cosh__def,axiom,
    ! [X: real] : aa_real_real(cosh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_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(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% cosh_def
tff(fact_8372_cosh__def,axiom,
    ! [X: complex] : aa_complex_complex(cosh_complex,X) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% cosh_def
tff(fact_8373_sinh__def,axiom,
    ! [X: real] : aa_real_real(sinh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% sinh_def
tff(fact_8374_sinh__def,axiom,
    ! [X: complex] : aa_complex_complex(sinh_complex,X) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% sinh_def
tff(fact_8375_exp__first__term,axiom,
    ! [X: real] : aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),suminf_real(aTP_Lamm_el(X))) ).

% exp_first_term
tff(fact_8376_exp__first__term,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) = aa_complex_complex(plus_plus_complex(one_one_complex),suminf_complex(aTP_Lamm_em(X))) ).

% exp_first_term
tff(fact_8377_cosh__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_en(X)),aa_real_real(cosh_real,X)) ).

% cosh_converges
tff(fact_8378_cosh__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_eo(X)),aa_complex_complex(cosh_complex,X)) ).

% cosh_converges
tff(fact_8379_sinh__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aTP_Lamm_ep(X)),aa_real_real(sinh_real,X)) ).

% sinh_converges
tff(fact_8380_sinh__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aTP_Lamm_eq(X)),aa_complex_complex(sinh_complex,X)) ).

% sinh_converges
tff(fact_8381_mono__SucI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,aa_nat_nat(suc,N)))
     => topolo7278393974255667507et_nat(X6) ) ).

% mono_SucI1
tff(fact_8382_mono__SucI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,aa_nat_nat(suc,N)))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI1
tff(fact_8383_mono__SucI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(X6,N)),aa_nat_num(X6,aa_nat_nat(suc,N)))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI1
tff(fact_8384_mono__SucI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,aa_nat_nat(suc,N)))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI1
tff(fact_8385_mono__SucI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,aa_nat_nat(suc,N)))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI1
tff(fact_8386_mono__SucI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,aa_nat_nat(suc,N)))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI1
tff(fact_8387_mono__SucI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N))),aa_nat_set_nat(X6,N))
     => topolo7278393974255667507et_nat(X6) ) ).

% mono_SucI2
tff(fact_8388_mono__SucI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N))),aa_nat_rat(X6,N))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI2
tff(fact_8389_mono__SucI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(X6,aa_nat_nat(suc,N))),aa_nat_num(X6,N))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI2
tff(fact_8390_mono__SucI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N))),aa_nat_nat(X6,N))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI2
tff(fact_8391_mono__SucI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N))),aa_nat_int(X6,N))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI2
tff(fact_8392_mono__SucI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N))),aa_nat_real(X6,N))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI2
tff(fact_8393_monoseq__Suc,axiom,
    ! [X6: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X6)
    <=> ( ! [N4: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N4)),aa_nat_set_nat(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N4))),aa_nat_set_nat(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8394_monoseq__Suc,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N4)),aa_nat_rat(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N4))),aa_nat_rat(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8395_monoseq__Suc,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [N4: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(X6,N4)),aa_nat_num(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(X6,aa_nat_nat(suc,N4))),aa_nat_num(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8396_monoseq__Suc,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N4)),aa_nat_nat(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N4))),aa_nat_nat(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8397_monoseq__Suc,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N4)),aa_nat_int(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N4))),aa_nat_int(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8398_monoseq__Suc,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N4)),aa_nat_real(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N4))),aa_nat_real(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8399_monoseq__def,axiom,
    ! [X6: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M5)),aa_nat_set_nat(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N4)),aa_nat_set_nat(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8400_monoseq__def,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M5)),aa_nat_rat(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N4)),aa_nat_rat(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8401_monoseq__def,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_num_o(ord_less_eq_num(aa_nat_num(X6,M5)),aa_nat_num(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_num_o(ord_less_eq_num(aa_nat_num(X6,N4)),aa_nat_num(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8402_monoseq__def,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M5)),aa_nat_nat(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N4)),aa_nat_nat(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8403_monoseq__def,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M5)),aa_nat_int(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N4)),aa_nat_int(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8404_monoseq__def,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M5)),aa_nat_real(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N4)),aa_nat_real(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8405_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_er),Na),zero_zero_real) ).

% of_nat_code
tff(fact_8406_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,Na) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_es,Na),zero_zero_int) ).

% of_nat_code
tff(fact_8407_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_et),Na),zero_zero_nat) ).

% of_nat_code
tff(fact_8408_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_complex(semiri8010041392384452111omplex,Na) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_eu,Na),zero_zero_complex) ).

% of_nat_code
tff(fact_8409_of__nat__code,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,Na) = semiri7787848453975740701ux_rat(aTP_Lamm_ev,Na,zero_zero_rat) ).

% of_nat_code
tff(fact_8410_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_ew(Z))) ).

% Arg_def
tff(fact_8411_set__vebt__def,axiom,
    ! [Tb: vEBT_VEBT] : vEBT_set_vebt(Tb) = collect_nat(vEBT_V8194947554948674370ptions(Tb)) ).

% set_vebt_def
tff(fact_8412_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ey(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y))) ).

% sin_x_sin_y
tff(fact_8413_sin__x__sin__y,axiom,
    ! [X: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_fa(X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% sin_x_sin_y
tff(fact_8414_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(ord_less_eq_set_nat(I),K) ) ).

% atMost_iff
tff(fact_8415_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( member_rat(I,set_ord_atMost_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K) ) ).

% atMost_iff
tff(fact_8416_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_atMost_num(K))
    <=> aa_num_o(ord_less_eq_num(I),K) ) ).

% atMost_iff
tff(fact_8417_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( member_int(I,set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K) ) ).

% atMost_iff
tff(fact_8418_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_ord_atMost_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K) ) ).

% atMost_iff
tff(fact_8419_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ).

% atMost_iff
tff(fact_8420_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_int(aa_complex_int(F,X2),ring_1_Ints_int) )
     => member_int(groups5690904116761175830ex_int(F,A2),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_8421_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_complex(aa_real_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(groups5754745047067104278omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8422_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_int(aa_real_int(F,X2),ring_1_Ints_int) )
     => member_int(groups1932886352136224148al_int(F,A2),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_8423_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => member_complex(aa_nat_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(aa_set_nat_complex(groups2073611262835488442omplex(F),A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8424_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => member_int(aa_nat_int(F,X2),ring_1_Ints_int) )
     => member_int(groups3539618377306564664at_int(F,A2),ring_1_Ints_int) ) ).

% Ints_sum
tff(fact_8425_Ints__sum,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => member_complex(aa_int_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(groups3049146728041665814omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8426_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_real(aa_complex_real(F,X2),ring_1_Ints_real) )
     => member_real(groups5808333547571424918x_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_8427_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_real(aa_real_real(F,X2),ring_1_Ints_real) )
     => member_real(groups8097168146408367636l_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_8428_Ints__sum,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => member_real(aa_int_real(F,X2),ring_1_Ints_real) )
     => member_real(groups8778361861064173332t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_8429_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_complex(aa_complex_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(groups7754918857620584856omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_sum
tff(fact_8430_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_fb(F),A2) ).

% of_nat_sum
tff(fact_8431_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_fc(F),A2) ).

% of_nat_sum
tff(fact_8432_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_fd(F),A2) ).

% of_nat_sum
tff(fact_8433_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_fe(F)),A2) ).

% of_nat_sum
tff(fact_8434_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups2906978787729119204at_rat(aTP_Lamm_ff(F),A2) ).

% of_nat_sum
tff(fact_8435_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fg(F)),A2) ).

% of_nat_sum
tff(fact_8436_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fh(F)),A2) ).

% of_nat_sum
tff(fact_8437_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_fi(F),A2) ).

% of_int_sum
tff(fact_8438_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fj(F)),A2) ).

% of_int_sum
tff(fact_8439_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_fk(F),A2) ).

% of_int_sum
tff(fact_8440_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_fl(F),A2) ).

% of_int_sum
tff(fact_8441_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_fm(F),A2) ).

% of_int_sum
tff(fact_8442_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X),set_or4236626031148496127et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(X),Y) ) ).

% atMost_subset_iff
tff(fact_8443_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(X),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% atMost_subset_iff
tff(fact_8444_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(X),set_ord_atMost_num(Y))
    <=> aa_num_o(ord_less_eq_num(X),Y) ) ).

% atMost_subset_iff
tff(fact_8445_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(X),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% atMost_subset_iff
tff(fact_8446_atMost__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(X),set_ord_atMost_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% atMost_subset_iff
tff(fact_8447_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% atMost_subset_iff
tff(fact_8448_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8449_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8450_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8451_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8452_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fn,M)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8453_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fo(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_8454_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_fp(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_8455_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8456_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8457_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = 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_da(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8458_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8459_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_fr(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8460_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_fs(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_8461_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ft(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_8462_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Na: nat,D: fun_nat_complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_complex(C,I3) = aa_nat_complex(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8463_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Na: nat,D: fun_nat_real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_real(C,I3) = aa_nat_real(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8464_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B2: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B2)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_8465_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_8466_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B2: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_8467_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_fu(U)) ).

% atMost_def
tff(fact_8468_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_fv,U)) ).

% atMost_def
tff(fact_8469_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aTP_Lamm_fw(U)) ).

% atMost_def
tff(fact_8470_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_fx,U)) ).

% atMost_def
tff(fact_8471_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_fy,U)) ).

% atMost_def
tff(fact_8472_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fz,U)) ).

% atMost_def
tff(fact_8473_sum__choose__lower,axiom,
    ! [R4: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ga,R4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R4),Na))),Na) ).

% sum_choose_lower
tff(fact_8474_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gb,Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8475_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gb,Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8476_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Na: nat,K: nat] :
      ( ! [W2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8477_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Na: nat,K: nat] :
      ( ! [W2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8478_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Na: nat] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_complex(C,I3) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8479_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Na: nat] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_real(C,I3) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8480_gbinomial__parallel__sum,axiom,
    ! [A: complex,Na: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_gc(A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Na))),one_one_complex)),Na) ).

% gbinomial_parallel_sum
tff(fact_8481_gbinomial__parallel__sum,axiom,
    ! [A: rat,Na: nat] : groups2906978787729119204at_rat(aTP_Lamm_gd(A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Na))),one_one_rat)),Na) ).

% gbinomial_parallel_sum
tff(fact_8482_gbinomial__parallel__sum,axiom,
    ! [A: real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ge(A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_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,Na))),one_one_real)),Na) ).

% gbinomial_parallel_sum
tff(fact_8483_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gf(M),Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8484_vandermonde,axiom,
    ! [M: nat,Na: nat,R4: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gg(M),Na),R4)),aa_nat_set_nat(set_ord_atMost_nat,R4)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),R4) ).

% vandermonde
tff(fact_8485_sum__gp__basic,axiom,
    ! [X: complex,Na: 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,Na))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8486_sum__gp__basic,axiom,
    ! [X: rat,Na: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,Na))) = minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8487_sum__gp__basic,axiom,
    ! [X: int,Na: 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,Na))) = 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,Na))) ).

% sum_gp_basic
tff(fact_8488_sum__gp__basic,axiom,
    ! [X: real,Na: 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(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8489_choose__row__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% choose_row_sum
tff(fact_8490_binomial,axiom,
    ! [A: nat,B: nat,Na: 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)),Na) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gh(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial
tff(fact_8491_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Na: 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))),Na)))) = groups2906978787729119204at_rat(aTP_Lamm_gi(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8492_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Na: 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))),Na)))) = groups3539618377306564664at_int(aTP_Lamm_gj(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8493_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Na: 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))),Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gk(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8494_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Na: 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))),Na)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gl(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8495_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_complex,Na: nat,B: fun_nat_complex,X: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(B),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_gn(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8496_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_rat,Na: nat,B: fun_nat_rat,X: rat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_rat(A,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(B),X),aa_nat_set_nat(set_ord_atMost_nat,Na))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_gq(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8497_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_int,Na: nat,B: fun_nat_int,X: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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_gr(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(B),X),aa_nat_set_nat(set_ord_atMost_nat,Na))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_gt(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8498_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_real,Na: nat,B: fun_nat_real,X: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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_bo(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(B),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_gv(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8499_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_gw(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8500_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_gx(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),M)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8501_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gy(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),M)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8502_binomial__ring,axiom,
    ! [A: int,B: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Na) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_gz(A),B),Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial_ring
tff(fact_8503_binomial__ring,axiom,
    ! [A: complex,B: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),Na) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ha(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial_ring
tff(fact_8504_binomial__ring,axiom,
    ! [A: rat,B: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Na) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_hb(A),B),Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial_ring
tff(fact_8505_binomial__ring,axiom,
    ! [A: nat,B: nat,Na: 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)),Na) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gh(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial_ring
tff(fact_8506_binomial__ring,axiom,
    ! [A: real,B: real,Na: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Na) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hc(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% binomial_ring
tff(fact_8507_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,Na: nat,B: fun_nat_nat,X: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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_hd(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hd(B),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_hf(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_8508_choose__square__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hg,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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))),Na)),Na) ).

% choose_square_sum
tff(fact_8509_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Na) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_hh(A),B),Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% pochhammer_binomial_sum
tff(fact_8510_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B)),Na) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_hi(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% pochhammer_binomial_sum
tff(fact_8511_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Na) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_hj(A),B),Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% pochhammer_binomial_sum
tff(fact_8512_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Na) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hk(A),B),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% pochhammer_binomial_sum
tff(fact_8513_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_hl(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_8514_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_hm(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_8515_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_hn(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_8516_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ho(X),M)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),I5)) ).

% sum_power_add
tff(fact_8517_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_hp(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_hq(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_8518_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_hr(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_hs(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_8519_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ht(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_hu(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_8520_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_hv(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_hw(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_8521_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_hx(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_hy(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_8522_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_hz(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ia(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8523_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ib(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ic(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8524_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_id(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ie(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8525_exp__series__add__commuting,axiom,
    ! [X: complex,Y: complex,Na: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),Na)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_if(X),Y),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ) ) ).

% exp_series_add_commuting
tff(fact_8526_exp__series__add__commuting,axiom,
    ! [X: real,Y: real,Na: 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(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),Na)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ig(X),Y),Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ) ) ).

% exp_series_add_commuting
tff(fact_8527_root__polyfun,axiom,
    ! [Na: nat,Z: int,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_int(power_power_int(Z),Na) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_ih(Na),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8528_root__polyfun,axiom,
    ! [Na: nat,Z: complex,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_complex(power_power_complex(Z),Na) = A )
      <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ii(Na),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8529_root__polyfun,axiom,
    ! [Na: nat,Z: code_integer,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),Na) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_ij(Na),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8530_root__polyfun,axiom,
    ! [Na: nat,Z: rat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_rat(power_power_rat(Z),Na) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ik(Na),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8531_root__polyfun,axiom,
    ! [Na: nat,Z: real,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_real(power_power_real(Z),Na) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_il(Na),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8532_sum__gp0,axiom,
    ! [X: complex,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = $ite(X = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na)))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ).

% sum_gp0
tff(fact_8533_sum__gp0,axiom,
    ! [X: rat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,Na)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))),minus_minus_rat(one_one_rat,X))) ).

% sum_gp0
tff(fact_8534_sum__gp0,axiom,
    ! [X: real,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = $ite(X = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,Na)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))) ).

% sum_gp0
tff(fact_8535_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_im(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8536_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_in,Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8537_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_io(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8538_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_ip(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8539_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_iq,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8540_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ir(M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8541_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(aTP_Lamm_is(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8542_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_it,M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8543_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_hz(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_iu(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8544_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ib(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_iv(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8545_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_id(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_iw(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8546_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_8547_choose__linear__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ix,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),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,Na),one_one_nat))) ).

% choose_linear_sum
tff(fact_8548_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( groups7501900531339628137nteger(aTP_Lamm_iy(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8549_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_iz,Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8550_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ja(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8551_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( groups2906978787729119204at_rat(aTP_Lamm_jb(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8552_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jc,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8553_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ? [M8: real] :
        ! [Z3: complex] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Na)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,Na)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8554_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ? [M8: real] :
        ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Na)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,Na)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8555_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_8556_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_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(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_8557_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_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(power_power_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_8558_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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_nat_fun_nat_int(aTP_Lamm_jd,Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8559_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_je(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8560_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_jf(Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8561_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jg,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8562_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( 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_nat_fun_nat_int(aTP_Lamm_jh,Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8563_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ji(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8564_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_jj(Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8565_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jk,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8566_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_jl(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_8567_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_jm(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_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_8568_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jn(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)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_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_8569_mask__eq__sum__exp,axiom,
    ! [Na: 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))),Na)),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_o(aTP_Lamm_jo,Na))) ).

% mask_eq_sum_exp
tff(fact_8570_mask__eq__sum__exp,axiom,
    ! [Na: 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))),Na)),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_o(aTP_Lamm_jo,Na))) ).

% mask_eq_sum_exp
tff(fact_8571_mask__eq__sum__exp__nat,axiom,
    ! [Na: 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))),Na)),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_o(aTP_Lamm_jo,Na))) ).

% mask_eq_sum_exp_nat
tff(fact_8572_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jq(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ).

% cos_x_cos_y
tff(fact_8573_cos__x__cos__y,axiom,
    ! [X: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_js(X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ).

% cos_x_cos_y
tff(fact_8574_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ju(X),Y)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))) ).

% sums_cos_x_plus_y
tff(fact_8575_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jw(X),Y)),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ).

% sums_cos_x_plus_y
tff(fact_8576_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ct(F)),A2)) ).

% sum_abs_ge_zero
tff(fact_8577_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_jx(F),A2)) ).

% sum_abs_ge_zero
tff(fact_8578_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ct(F)),A2)) ).

% sum_abs
tff(fact_8579_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_jx(F),A2)) ).

% sum_abs
tff(fact_8580_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I2)) )
     => ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( member_complex(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_jy(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8581_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I2)) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_jz(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8582_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2)) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ka(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8583_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2)) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_kb(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8584_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X,I2)) )
     => ( ( groups5058264527183730370ex_rat(X,I5) = one_one_rat )
       => ( ! [I2: complex] :
              ( member_complex(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_complex_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kc(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8585_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I2)) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_real_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kd(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8586_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I2)) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ke(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8587_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(X,I2)) )
     => ( ( groups3906332499630173760nt_rat(X,I5) = one_one_rat )
       => ( ! [I2: int] :
              ( member_int(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_kf(X),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8588_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_int,A: fun_complex_int,B: int,Delta: int] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(X,I2)) )
     => ( ( groups5690904116761175830ex_int(X,I5) = one_one_int )
       => ( ! [I2: complex] :
              ( member_complex(I2,I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_complex_int(A,I2)),B))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_kg(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8589_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_int,A: fun_real_int,B: int,Delta: int] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(X,I2)) )
     => ( ( groups1932886352136224148al_int(X,I5) = one_one_int )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_int_o(aa_int_fun_int_o(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,I2)),B))),Delta) )
         => aa_int_o(aa_int_fun_int_o(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_kh(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8590_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_8591_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_ao,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_8592_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ce),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_8593_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_as,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_8594_of__nat__id,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Na) = Na ).

% of_nat_id
tff(fact_8595_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_fc(F),A2) ).

% int_sum
tff(fact_8596_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_fd(F),A2) ).

% int_sum
tff(fact_8597_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X2)),aa_complex_nat(F,X2)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ki(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8598_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X2)),aa_real_nat(F,X2)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kj(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8599_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X2: set_nat] :
          ( member_set_nat(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X2)),aa_set_nat_nat(F,X2)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kk(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8600_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X2)),aa_int_nat(F,X2)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kl(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8601_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X2)),aa_nat_nat(F,X2)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_km(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_8602_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Na: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,Na) )
     => ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2)) ) ) ).

% sum_SucD
tff(fact_8603_sum__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_aj,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_kn(Na),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8604_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_aj,collect_complex(aTP_Lamm_ko(Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8605_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: real] :
            ( member_real(A4,A2)
           => ( aa_real_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8606_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),A2) != zero_zero_complex )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8607_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: int] :
            ( member_int(A4,A2)
           => ( aa_int_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8608_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups5808333547571424918x_real(G,A2) != zero_zero_real )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A2)
           => ( aa_complex_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8609_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A4: real] :
            ( member_real(A4,A2)
           => ( aa_real_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8610_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A4: int] :
            ( member_int(A4,A2)
           => ( aa_int_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8611_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups5058264527183730370ex_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A2)
           => ( aa_complex_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8612_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: real] :
            ( member_real(A4,A2)
           => ( aa_real_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8613_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8614_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: int] :
            ( member_int(A4,A2)
           => ( aa_int_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8615_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_nat_nat(G,X2) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_8616_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => ( aa_complex_complex(G,X2) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_8617_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_nat_real(G,X2) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_8618_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => ( aa_int_int(G,X2) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_8619_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).

% sum_mono
tff(fact_8620_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_8621_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_8622_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_8623_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).

% sum_mono
tff(fact_8624_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_8625_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_8626_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I2: complex] :
          ( member_complex(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).

% sum_mono
tff(fact_8627_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_8628_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_8629_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ck(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_8630_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_kp(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_8631_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(G),H)),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),A2)),aa_set_nat_real(groups6591440286371151544t_real(H),A2)) ).

% sum.distrib
tff(fact_8632_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kq(G),H),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_8633_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R4: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A2)),R4) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_kr(F),R4),A2) ).

% sum_divide_distrib
tff(fact_8634_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A2: set_nat,R4: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),R4) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bw(F),R4)),A2) ).

% sum_divide_distrib
tff(fact_8635_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8636_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8637_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8638_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8639_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_8640_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_8641_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_8642_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X2)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A2)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_8643_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X2)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_8644_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A2)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_8645_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_8646_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_8647_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_8648_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_8649_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_8650_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_8651_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_8652_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups5690904116761175830ex_int(F,A2)) ) ).

% sum_nonneg
tff(fact_8653_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A2)) ) ).

% sum_nonneg
tff(fact_8654_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups3539618377306564664at_int(F,A2)) ) ).

% sum_nonneg
tff(fact_8655_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A2)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X2)) = aa_nat_nat(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8656_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A2)
           => ( aa_nat_real(F,aa_nat_nat(suc,X2)) = aa_nat_real(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8657_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),zero_zero_real)
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ks(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8658_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),X)
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ks(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8659_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),X)
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kt(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8660_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X)
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kt(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8661_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(ord_less_set_nat(I),K) ) ).

% lessThan_iff
tff(fact_8662_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( member_rat(I,set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K) ) ).

% lessThan_iff
tff(fact_8663_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_lessThan_num(K))
    <=> aa_num_o(ord_less_num(I),K) ) ).

% lessThan_iff
tff(fact_8664_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( member_int(I,set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).

% lessThan_iff
tff(fact_8665_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% lessThan_iff
tff(fact_8666_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K) ) ).

% lessThan_iff
tff(fact_8667_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(X),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8668_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(X),set_ord_lessThan_num(Y))
    <=> aa_num_o(ord_less_eq_num(X),Y) ) ).

% lessThan_subset_iff
tff(fact_8669_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ord_less_eq_set_int(set_ord_lessThan_int(X),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8670_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8671_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ord_less_eq_set_real(set_or5984915006950818249n_real(X),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8672_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_rat(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8673_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_int(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8674_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_nat(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8675_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_real(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8676_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ku),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8677_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_kv(U)) ).

% lessThan_def
tff(fact_8678_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_kw,U)) ).

% lessThan_def
tff(fact_8679_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aTP_Lamm_kx(U)) ).

% lessThan_def
tff(fact_8680_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_ky,U)) ).

% lessThan_def
tff(fact_8681_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jo,U)) ).

% lessThan_def
tff(fact_8682_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_kz,U)) ).

% lessThan_def
tff(fact_8683_lessThan__strict__subset__iff,axiom,
    ! [M: rat,Na: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(M),set_ord_lessThan_rat(Na))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8684_lessThan__strict__subset__iff,axiom,
    ! [M: num,Na: num] :
      ( ord_less_set_num(set_ord_lessThan_num(M),set_ord_lessThan_num(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8685_lessThan__strict__subset__iff,axiom,
    ! [M: int,Na: int] :
      ( ord_less_set_int(set_ord_lessThan_int(M),set_ord_lessThan_int(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8686_lessThan__strict__subset__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8687_lessThan__strict__subset__iff,axiom,
    ! [M: real,Na: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(M),set_or5984915006950818249n_real(Na))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8688_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_8689_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(A),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8690_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> aa_num_o(ord_less_num(A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8691_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(A),set_ord_lessThan_int(B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8692_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8693_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(A),set_or5984915006950818249n_real(B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8694_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_la(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nat_diff_reindex
tff(fact_8695_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lb(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nat_diff_reindex
tff(fact_8696_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Na: real] :
      ( ! [X2: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X2)),aa_real_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Na))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Na))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kj(Q),P),set_or5984915006950818249n_real(Na)) ) ) ).

% sum_diff_distrib
tff(fact_8697_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Na: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X2)),aa_nat_nat(P,X2))
     => ( 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,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_km(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ) ) ).

% sum_diff_distrib
tff(fact_8698_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),X) ) ) ).

% suminf_le_const
tff(fact_8699_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),X) ) ) ).

% suminf_le_const
tff(fact_8700_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),X) ) ) ).

% suminf_le_const
tff(fact_8701_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8702_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8703_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = 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_da(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8704_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8705_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_fr(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8706_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_fs(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_8707_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ft(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_8708_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_lc(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8709_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_ld(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_8710_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_le(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_8711_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8712_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8713_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8714_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Na: nat,S: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),Na)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na)))) ) ).

% sums_iff_shift
tff(fact_8715_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Na: nat,S: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),Na)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))))
    <=> aa_real_o(sums_real(F),S) ) ).

% sums_iff_shift'
tff(fact_8716_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,Na: nat] :
      ( aa_real_o(sums_real(F),S)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),Na)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na)))) ) ).

% sums_split_initial_segment
tff(fact_8717_one__diff__power__eq,axiom,
    ! [X: complex,Na: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),Na)) = 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,Na))) ).

% one_diff_power_eq
tff(fact_8718_one__diff__power__eq,axiom,
    ! [X: rat,Na: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),Na)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq
tff(fact_8719_one__diff__power__eq,axiom,
    ! [X: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),Na)) = 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,Na))) ).

% one_diff_power_eq
tff(fact_8720_one__diff__power__eq,axiom,
    ! [X: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),Na)) = 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(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq
tff(fact_8721_power__diff__1__eq,axiom,
    ! [X: complex,Na: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),Na)),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,Na))) ).

% power_diff_1_eq
tff(fact_8722_power__diff__1__eq,axiom,
    ! [X: rat,Na: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),Na),one_one_rat) = aa_rat_rat(times_times_rat(minus_minus_rat(X,one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_1_eq
tff(fact_8723_power__diff__1__eq,axiom,
    ! [X: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),Na)),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,Na))) ).

% power_diff_1_eq
tff(fact_8724_power__diff__1__eq,axiom,
    ! [X: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),Na)),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(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_1_eq
tff(fact_8725_geometric__sum,axiom,
    ! [X: complex,Na: nat] :
      ( ( X != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),Na)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_8726_geometric__sum,axiom,
    ! [X: rat,Na: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),Na),one_one_rat),minus_minus_rat(X,one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8727_geometric__sum,axiom,
    ! [X: real,Na: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),Na)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8728_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8729_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8730_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_da(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8731_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8732_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_8733_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_cg(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_8734_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Na: nat] :
      ( summable_int(F)
     => ( ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M4)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M4)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_8735_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( summable_nat(F)
     => ( ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M4)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_8736_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Na: nat] :
      ( summable_real(F)
     => ( ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M4)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M4)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_8737_sum__gp__strict,axiom,
    ! [X: complex,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = $ite(X = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Na),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),Na))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ).

% sum_gp_strict
tff(fact_8738_sum__gp__strict,axiom,
    ! [X: rat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Na),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),Na)),minus_minus_rat(one_one_rat,X))) ).

% sum_gp_strict
tff(fact_8739_sum__gp__strict,axiom,
    ! [X: real,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = $ite(X = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Na),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),Na))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))) ).

% sum_gp_strict
tff(fact_8740_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lf(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lg(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8741_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lh(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_li(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8742_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lj(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lk(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8743_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_ll(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_lm(Z),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8744_power__diff__sumr2,axiom,
    ! [X: complex,Na: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),Na)),aa_nat_complex(power_power_complex(Y),Na)) = 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_ln(X),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8745_power__diff__sumr2,axiom,
    ! [X: rat,Na: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),Na),aa_nat_rat(power_power_rat(Y),Na)) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_lo(X),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8746_power__diff__sumr2,axiom,
    ! [X: int,Na: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),Na)),aa_nat_int(power_power_int(Y),Na)) = 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_lp(X),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8747_power__diff__sumr2,axiom,
    ! [X: real,Na: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),Na)),aa_nat_real(power_power_real(Y),Na)) = 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_lq(X),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8748_diff__power__eq__sum,axiom,
    ! [X: complex,Na: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,Na))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Na))) = 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_lr(X),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8749_diff__power__eq__sum,axiom,
    ! [X: rat,Na: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na)),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ls(X),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8750_diff__power__eq__sum,axiom,
    ! [X: int,Na: 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,Na))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Na))) = 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_lt(X),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8751_diff__power__eq__sum,axiom,
    ! [X: real,Na: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),aa_nat_nat(suc,Na))),aa_nat_real(power_power_real(Y),aa_nat_nat(suc,Na))) = 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_lu(X),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8752_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,Na: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex )
     => ~ ! [B4: fun_nat_complex] :
            ~ ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(B4),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8753_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,Na: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(C),A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat )
     => ~ ! [B4: fun_nat_rat] :
            ~ ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(times_times_rat(minus_minus_rat(Z3,A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(B4),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8754_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,Na: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(C),A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int )
     => ~ ! [B4: fun_nat_int] :
            ~ ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_gr(B4),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8755_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,Na: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real )
     => ~ ! [B4: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_bo(B4),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8756_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,Na: nat,A: complex] :
    ? [B4: fun_nat_complex] :
    ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_bq(B4),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Na)))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% polyfun_linear_factor
tff(fact_8757_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,Na: nat,A: rat] :
    ? [B4: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(minus_minus_rat(Z3,A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(B4),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Na)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(C),A),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% polyfun_linear_factor
tff(fact_8758_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,Na: nat,A: int] :
    ? [B4: fun_nat_int] :
    ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_gr(B4),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Na)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(C),A),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% polyfun_linear_factor
tff(fact_8759_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,Na: nat,A: real] :
    ? [B4: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_bo(B4),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Na)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% polyfun_linear_factor
tff(fact_8760_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_rat,K5: rat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K5) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
       => aa_rat_o(aa_rat_fun_rat_o(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,Na),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8761_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
       => aa_int_o(aa_int_fun_int_o(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,Na),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8762_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
       => aa_nat_o(aa_nat_fun_nat_o(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,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8763_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K5) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
       => aa_real_o(aa_real_fun_real_o(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,Na),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8764_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Na: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M4)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),I)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8765_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Na: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M4)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),I)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8766_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Na: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M4)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),I)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8767_one__diff__power__eq_H,axiom,
    ! [X: complex,Na: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),Na)) = 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_lv(X),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8768_one__diff__power__eq_H,axiom,
    ! [X: rat,Na: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),Na)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_lw(X),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8769_one__diff__power__eq_H,axiom,
    ! [X: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),Na)) = 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_lx(X),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8770_one__diff__power__eq_H,axiom,
    ! [X: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),Na)) = 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_ly(X),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8771_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_lz(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8772_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ma(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8773_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_mb(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8774_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_mc(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8775_Maclaurin__zero,axiom,
    ! [X: real,Na: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_md(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8776_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(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_me(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B6),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(H),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_8777_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mf(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))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mg(F)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum_split_even_odd
tff(fact_8778_Maclaurin__exp__le,axiom,
    ! [X: real,Na: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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(aTP_Lamm_mi(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_8779_polyfun__diff__alt,axiom,
    ! [Na: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = 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_mk(Na),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8780_polyfun__diff__alt,axiom,
    ! [Na: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(A),X),aa_nat_set_nat(set_ord_atMost_nat,Na)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_mm(Na),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8781_polyfun__diff__alt,axiom,
    ! [Na: nat,A: fun_nat_int,X: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(A),X),aa_nat_set_nat(set_ord_atMost_nat,Na))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = 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_mo(Na),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8782_polyfun__diff__alt,axiom,
    ! [Na: nat,A: fun_nat_real,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = 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_mq(Na),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8783_exp__first__terms,axiom,
    ! [X: complex,K: nat] : aa_complex_complex(exp_complex,X) = aa_complex_complex(plus_plus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_du(X)),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mr(X),K))) ).

% exp_first_terms
tff(fact_8784_exp__first__terms,axiom,
    ! [X: real,K: nat] : aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dt(X)),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ms(X),K))) ).

% exp_first_terms
tff(fact_8785_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(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_8786_Maclaurin__exp__lt,axiom,
    ! [X: real,Na: nat] :
      ( ( X != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T)),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(aTP_Lamm_mi(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8787_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Na: 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)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_mu(H),Z),Na)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8788_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Na: nat] :
      ( ( H != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),H)),Na),aa_nat_rat(power_power_rat(Z),Na)),H),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_mw(H),Z),Na),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8789_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Na: 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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(power_power_real(Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_my(H),Z),Na)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8790_Maclaurin__sin__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T: real] : aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kt(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ).

% Maclaurin_sin_expansion
tff(fact_8791_Maclaurin__sin__expansion2,axiom,
    ! [X: real,Na: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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(aTP_Lamm_kt(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8792_Maclaurin__cos__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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(aTP_Lamm_ks(X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8793_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => bij_betw_nat_complex(aTP_Lamm_mz(Na),aa_nat_set_nat(set_ord_lessThan_nat,Na),collect_complex(aTP_Lamm_ko(Na))) ) ).

% bij_betw_roots_unity
tff(fact_8794_sum__gp,axiom,
    ! [X: complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),
        zero_zero_complex,
        $ite(X = one_one_complex,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,Na),one_one_nat)),M)),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,Na)))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ) ).

% sum_gp
tff(fact_8795_sum__gp,axiom,
    ! [X: rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),
        zero_zero_rat,
        $ite(X = one_one_rat,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,Na),one_one_nat)),M)),divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))),minus_minus_rat(one_one_rat,X))) ) ).

% sum_gp
tff(fact_8796_sum__gp,axiom,
    ! [X: real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),
        zero_zero_real,
        $ite(X = one_one_real,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,Na),one_one_nat)),M)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,Na)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))) ) ).

% sum_gp
tff(fact_8797_gchoose__row__sum__weighted,axiom,
    ! [R4: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_jl(R4)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R4),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8798_gchoose__row__sum__weighted,axiom,
    ! [R4: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_jm(R4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(times_times_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R4),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8799_gchoose__row__sum__weighted,axiom,
    ! [R4: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jn(R4)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(R4),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_8800_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8801_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = 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,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8802_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L4: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L4,H2) )
    <=> ( ( ( L = L4 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
          & ~ aa_set_nat_o(ord_less_eq_set_nat(L4),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8803_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L4: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L4,H2) )
    <=> ( ( ( L = L4 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L4),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8804_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L4: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L4,H2) )
    <=> ( ( ( L = L4 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(ord_less_eq_num(L),H)
          & ~ aa_num_o(ord_less_eq_num(L4),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8805_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L4: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L4),H2) )
    <=> ( ( ( L = L4 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L4),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8806_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L4: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L4,H2) )
    <=> ( ( ( L = L4 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L4),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8807_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L4: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L4,H2) )
    <=> ( ( ( L = L4 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L4),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8808_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8809_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( member_rat(I,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8810_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(ord_less_eq_num(L),I)
        & aa_num_o(ord_less_eq_num(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8811_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8812_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( member_int(I,set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8813_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( member_real(I,set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8814_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
          & aa_set_nat_o(ord_less_eq_set_nat(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8815_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8816_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(ord_less_eq_num(A),B)
        | ( aa_num_o(ord_less_eq_num(C),A)
          & aa_num_o(ord_less_eq_num(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8817_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8818_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8819_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8820_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))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
        | aa_set_nat_o(ord_less_eq_set_nat(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8821_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H),set_ord_atMost_rat(H2))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8822_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(L,H),set_ord_atMost_num(H2))
    <=> ( ~ aa_num_o(ord_less_eq_num(L),H)
        | aa_num_o(ord_less_eq_num(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8823_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8824_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(L,H),set_ord_atMost_int(H2))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8825_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(L,H),set_ord_atMost_real(H2))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8826_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_complex,aa_complex_complex(plus_plus_complex(aa_set_nat_complex(groups2073611262835488442omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_complex(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_8827_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_rat(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_8828_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] :
      groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_int(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_8829_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_nat,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),Na))),aa_nat_nat(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_8830_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_real(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_8831_all__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Na)
         => aa_nat_o(P,M5) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less
tff(fact_8832_ex__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Na)
          & aa_nat_o(P,M5) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less
tff(fact_8833_atMost__atLeast0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,Na) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na) ).

% atMost_atLeast0
tff(fact_8834_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8835_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8836_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_na(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8837_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,Na: nat] : 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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_cg(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8838_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
            & aa_set_nat_o(ord_less_eq_set_nat(B),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A)
              | aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8839_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8840_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(ord_less_eq_num(A),B)
          | ( aa_num_o(ord_less_eq_num(C),A)
            & aa_num_o(ord_less_eq_num(B),D)
            & ( aa_num_o(ord_less_num(C),A)
              | aa_num_o(ord_less_num(B),D) ) ) )
        & aa_num_o(ord_less_eq_num(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8841_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8842_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8843_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8844_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nb(G),Na),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8845_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Na: nat,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_nc(G),Na),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8846_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8847_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_8848_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_8849_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_8850_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_8851_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8852_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8853_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = 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),Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8854_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = 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),Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8855_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = aa_rat_rat(aa_rat_fun_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)),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8856_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8857_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8858_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8859_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,aa_nat_nat(suc,Na))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8860_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Na))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8861_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8862_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Na))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8863_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8864_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8865_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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),Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8866_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8867_sum__Suc__diff,axiom,
    ! [M: nat,Na: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups2906978787729119204at_rat(aTP_Lamm_lc(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Na)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8868_sum__Suc__diff,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups3539618377306564664at_int(aTP_Lamm_ld(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Na))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8869_sum__Suc__diff,axiom,
    ! [M: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_le(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Na))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8870_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.atLeast1_atMost_eq
tff(fact_8871_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.atLeast1_atMost_eq
tff(fact_8872_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_da(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8873_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8874_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nd(A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nf(A),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nested_swap'
tff(fact_8875_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ng(A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ni(A),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nested_swap'
tff(fact_8876_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_nj(F),A,B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_8877_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_nk(F),A,B,zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8878_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_nl(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8879_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_nm(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8880_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_nn(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8881_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),P3))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8882_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8883_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),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),Na))),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,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8884_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),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),Na))),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,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8885_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,M: nat,Na: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8886_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M: nat,Na: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8887_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8888_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8889_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_no(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,M)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_8890_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_np(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_8891_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,M: nat,Na: nat] :
      groups3539618377306564664at_int(aTP_Lamm_nq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),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,Na),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_8892_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nr(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),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,Na),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_8893_sum__telescope_H_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups2906978787729119204at_rat(aTP_Lamm_ns(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) = minus_minus_rat(aa_nat_rat(F,Na),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_8894_sum__telescope_H_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups3539618377306564664at_int(aTP_Lamm_nt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Na)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_8895_sum__telescope_H_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nu(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Na)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_8896_sum__power__shift,axiom,
    ! [M: nat,Na: nat,X: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8897_sum__power__shift,axiom,
    ! [M: nat,Na: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8898_sum__power__shift,axiom,
    ! [M: nat,Na: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8899_sum__power__shift,axiom,
    ! [M: nat,Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8900_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N7)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8901_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N7)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8902_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,X: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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),Na))) = 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,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_8903_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) = minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_8904_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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),Na))) = 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,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_8905_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_8906_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,Na: 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))),Na)))) = groups2906978787729119204at_rat(aTP_Lamm_gi(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.in_pairs
tff(fact_8907_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,Na: 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))),Na)))) = groups3539618377306564664at_int(aTP_Lamm_gj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.in_pairs
tff(fact_8908_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,Na: 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))),Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.in_pairs
tff(fact_8909_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,Na: 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))),Na)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gl(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% sum.in_pairs
tff(fact_8910_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Na: nat,K: complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))
           => ( aa_nat_complex(C,X4) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8911_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Na: nat,K: real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))
           => ( aa_nat_real(C,X4) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8912_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_nv(K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8913_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : groups2906978787729119204at_rat(aTP_Lamm_nw(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8914_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nx,K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8915_gauss__sum__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8916_double__arith__series,axiom,
    ! [A: int,D: int,Na: 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_ny(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) = 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,Na)),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,Na)),D))) ).

% double_arith_series
tff(fact_8917_double__arith__series,axiom,
    ! [A: complex,D: complex,Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_nz(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),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,Na)),D))) ).

% double_arith_series
tff(fact_8918_double__arith__series,axiom,
    ! [A: rat,D: rat,Na: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_oa(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),D))) ).

% double_arith_series
tff(fact_8919_double__arith__series,axiom,
    ! [A: nat,D: nat,Na: 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_ob(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) = 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,Na)),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,Na)),D))) ).

% double_arith_series
tff(fact_8920_double__arith__series,axiom,
    ! [A: real,D: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oc(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),D))) ).

% double_arith_series
tff(fact_8921_double__gauss__sum,axiom,
    ! [Na: 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),Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)) ).

% double_gauss_sum
tff(fact_8922_double__gauss__sum,axiom,
    ! [Na: 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),Na))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)) ).

% double_gauss_sum
tff(fact_8923_double__gauss__sum,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)) ).

% double_gauss_sum
tff(fact_8924_double__gauss__sum,axiom,
    ! [Na: 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),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)) ).

% double_gauss_sum
tff(fact_8925_double__gauss__sum,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)) ).

% double_gauss_sum
tff(fact_8926_arith__series__nat,axiom,
    ! [A: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_od(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = 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,Na)),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,Na),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8927_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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_8928_gauss__sum,axiom,
    ! [Na: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8929_gauss__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = 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,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_8930_arith__series,axiom,
    ! [A: int,D: int,Na: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ny(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_int_int(aa_int_fun_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,Na)),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,Na)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8931_arith__series,axiom,
    ! [A: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ob(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = 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,Na)),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,Na)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_8932_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: 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)),Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8933_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: 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)),Na))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8934_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_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)),Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8935_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: 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)),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8936_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_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)),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8937_sum__gp__offset,axiom,
    ! [X: complex,M: nat,Na: nat] :
      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),Na))) = $ite(X = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex),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,Na))))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ).

% sum_gp_offset
tff(fact_8938_sum__gp__offset,axiom,
    ! [X: rat,M: nat,Na: nat] :
      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),Na))) = $ite(X = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat),divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Na)))),minus_minus_rat(one_one_rat,X))) ).

% sum_gp_offset
tff(fact_8939_sum__gp__offset,axiom,
    ! [X: real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_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),Na))) = $ite(X = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,Na))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))) ).

% sum_gp_offset
tff(fact_8940_polyfun__diff,axiom,
    ! [Na: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_bq(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = 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_of(Na),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8941_polyfun__diff,axiom,
    ! [Na: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(A),X),aa_nat_set_nat(set_ord_atMost_nat,Na)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_oh(Na),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8942_polyfun__diff,axiom,
    ! [Na: nat,A: fun_nat_int,X: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(A),X),aa_nat_set_nat(set_ord_atMost_nat,Na))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gr(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = 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_oj(Na),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8943_polyfun__diff,axiom,
    ! [Na: nat,A: fun_nat_real,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_bo(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = 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_ol(Na),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8944_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,Na))),aa_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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_nat_nat(suc,Na))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_om(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))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8945_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Na))),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,Na))) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_on(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))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8946_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Na))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,Na))) = groups73079841787564623at_rat(aTP_Lamm_oo(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))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8947_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          $let(
                            half: nat,
                            half:= aa_nat_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(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))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_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(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,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_8948_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = produc4245557441103728435nt_int(aTP_Lamm_op,unique5052692396658037445od_int(M,Na)) ).

% divmod_algorithm_code(6)
tff(fact_8949_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_oq),unique5055182867167087721od_nat(M,Na)) ).

% divmod_algorithm_code(6)
tff(fact_8950_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = produc6916734918728496179nteger(aTP_Lamm_or,unique3479559517661332726nteger(M,Na)) ).

% divmod_algorithm_code(6)
tff(fact_8951_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_os,Y)) ).

% arctan_def
tff(fact_8952_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_complex(aa_complex_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(groups3708469109370488835omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8953_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_int(aa_complex_int(F,X2),ring_1_Ints_int) )
     => member_int(groups858564598930262913ex_int(F,A2),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_8954_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_complex(aa_real_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(groups713298508707869441omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8955_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_int(aa_real_int(F,X2),ring_1_Ints_int) )
     => member_int(groups4694064378042380927al_int(F,A2),ring_1_Ints_int) ) ).

% Ints_prod
tff(fact_8956_Ints__prod,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => member_complex(aa_nat_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(aa_set_nat_complex(groups6464643781859351333omplex(F),A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8957_Ints__prod,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => member_complex(aa_int_complex(F,X2),ring_1_Ints_complex) )
     => member_complex(groups7440179247065528705omplex(F,A2),ring_1_Ints_complex) ) ).

% Ints_prod
tff(fact_8958_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_real(aa_complex_real(F,X2),ring_1_Ints_real) )
     => member_real(groups766887009212190081x_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8959_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_real(aa_real_real(F,X2),ring_1_Ints_real) )
     => member_real(groups1681761925125756287l_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8960_Ints__prod,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => member_real(aa_nat_real(F,X2),ring_1_Ints_real) )
     => member_real(aa_set_nat_real(groups129246275422532515t_real(F),A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8961_Ints__prod,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => member_real(aa_int_real(F,X2),ring_1_Ints_real) )
     => member_real(groups2316167850115554303t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_8962_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ot),A2) = one_one_nat ).

% prod.neutral_const
tff(fact_8963_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_ou,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8964_prod_Oneutral__const,axiom,
    ! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ov,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8965_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_fc(F),A2) ).

% of_nat_prod
tff(fact_8966_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_fh(F)),A2) ).

% of_nat_prod
tff(fact_8967_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_fe(F)),A2) ).

% of_nat_prod
tff(fact_8968_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups73079841787564623at_rat(aTP_Lamm_ff(F),A2) ).

% of_nat_prod
tff(fact_8969_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fg(F)),A2) ).

% of_nat_prod
tff(fact_8970_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_fd(F),A2) ).

% of_nat_prod
tff(fact_8971_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_fj(F)),A2) ).

% of_int_prod
tff(fact_8972_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A2)) = groups73079841787564623at_rat(aTP_Lamm_ow(F),A2) ).

% of_int_prod
tff(fact_8973_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A2)) = groups705719431365010083at_int(aTP_Lamm_ox(F),A2) ).

% of_int_prod
tff(fact_8974_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_fk(F),A2) ).

% of_int_prod
tff(fact_8975_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_fl(F),A2) ).

% of_int_prod
tff(fact_8976_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_fm(F),A2) ).

% of_int_prod
tff(fact_8977_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_real(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8978_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_rat(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8979_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_nat(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8980_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_int(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8981_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8982_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8983_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8984_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8985_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_complex,aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_complex(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8986_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_real(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8987_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_rat(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8988_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_nat,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),Na))),aa_nat_nat(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8989_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_int(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8990_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,Na)) = produc4245557441103728435nt_int(aTP_Lamm_oy,unique5052692396658037445od_int(M,Na)) ).

% divmod_algorithm_code(5)
tff(fact_8991_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,Na)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_oz),unique5055182867167087721od_nat(M,Na)) ).

% divmod_algorithm_code(5)
tff(fact_8992_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,Na)) = produc6916734918728496179nteger(aTP_Lamm_pa,unique3479559517661332726nteger(M,Na)) ).

% divmod_algorithm_code(5)
tff(fact_8993_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),Na) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pb(F),Na)),A2) ).

% prod_power_distrib
tff(fact_8994_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A2: set_nat,Na: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A2)),Na) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pc(F),Na),A2) ).

% prod_power_distrib
tff(fact_8995_prod__power__distrib,axiom,
    ! [F: fun_int_int,A2: set_int,Na: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A2)),Na) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_pd(F),Na),A2) ).

% prod_power_distrib
tff(fact_8996_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_nat_nat(G,X2) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8997_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_nat_int(G,X2) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8998_prod_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => ( aa_int_int(G,X2) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8999_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_complex,A2: set_complex] :
      ( ( groups3708469109370488835omplex(G,A2) != one_one_complex )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A2)
           => ( aa_complex_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9000_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups713298508707869441omplex(G,A2) != one_one_complex )
     => ~ ! [A4: real] :
            ( member_real(A4,A2)
           => ( aa_real_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9001_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( aa_set_nat_complex(groups6464643781859351333omplex(G),A2) != one_one_complex )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9002_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups7440179247065528705omplex(G,A2) != one_one_complex )
     => ~ ! [A4: int] :
            ( member_int(A4,A2)
           => ( aa_int_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9003_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups766887009212190081x_real(G,A2) != one_one_real )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A2)
           => ( aa_complex_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9004_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups1681761925125756287l_real(G,A2) != one_one_real )
     => ~ ! [A4: real] :
            ( member_real(A4,A2)
           => ( aa_real_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9005_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A2: set_nat] :
      ( ( aa_set_nat_real(groups129246275422532515t_real(G),A2) != one_one_real )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9006_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups2316167850115554303t_real(G,A2) != one_one_real )
     => ~ ! [A4: int] :
            ( member_int(A4,A2)
           => ( aa_int_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9007_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups225925009352817453ex_rat(G,A2) != one_one_rat )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A2)
           => ( aa_complex_rat(G,A4) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9008_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups4061424788464935467al_rat(G,A2) != one_one_rat )
     => ~ ! [A4: real] :
            ( member_real(A4,A2)
           => ( aa_real_rat(G,A4) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_9009_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fo(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),A) ).

% mod_prod_eq
tff(fact_9010_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A2: set_nat] : aa_int_int(modulo_modulo_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_pe(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups705719431365010083at_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_9011_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_fp(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_9012_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2)) ) ).

% prod_mono
tff(fact_9013_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2)) ) ).

% prod_mono
tff(fact_9014_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),groups73079841787564623at_rat(G,A2)) ) ).

% prod_mono
tff(fact_9015_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(G,A2)) ) ).

% prod_mono
tff(fact_9016_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,A2)) ) ).

% prod_mono
tff(fact_9017_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2)) ) ).

% prod_mono
tff(fact_9018_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( member_int(I2,A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2)) ) ).

% prod_mono
tff(fact_9019_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,A2)) ) ).

% prod_mono
tff(fact_9020_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,A2)) ) ).

% prod_mono
tff(fact_9021_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2)) ) ).

% prod_mono
tff(fact_9022_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_nonneg
tff(fact_9023_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A2)) ) ).

% prod_nonneg
tff(fact_9024_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_nonneg
tff(fact_9025_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_pos
tff(fact_9026_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A2)) ) ).

% prod_pos
tff(fact_9027_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_pos
tff(fact_9028_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_9029_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_9030_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_9031_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_9032_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_complex_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups861055069439313189ex_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_9033_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_9034_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A2)) ) ).

% prod_ge_1
tff(fact_9035_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups858564598930262913ex_int(F,A2)) ) ).

% prod_ge_1
tff(fact_9036_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,X2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A2)) ) ).

% prod_ge_1
tff(fact_9037_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A2)) ) ).

% prod_ge_1
tff(fact_9038_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_9039_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))) = groups705719431365010083at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_9040_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A2: set_nat] : aa_nat_real(power_power_real(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_pf(C),F)),A2) ).

% power_sum
tff(fact_9041_power__sum,axiom,
    ! [C: complex,F: fun_nat_nat,A2: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_pg(C),F)),A2) ).

% power_sum
tff(fact_9042_power__sum,axiom,
    ! [C: rat,F: fun_nat_nat,A2: set_nat] : aa_nat_rat(power_power_rat(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups73079841787564623at_rat(aa_fun2283220954820195200at_rat(aTP_Lamm_ph(C),F),A2) ).

% power_sum
tff(fact_9043_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pi(C),F)),A2) ).

% power_sum
tff(fact_9044_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A2: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_pj(C),F),A2) ).

% power_sum
tff(fact_9045_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A2: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A2)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_pk(C),F),A2) ).

% power_sum
tff(fact_9046_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Na: nat] : 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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_na(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_9047_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pl(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_9048_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_9049_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_9050_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_9051_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_9052_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_9053_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_9054_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A2)),one_one_nat) ) ).

% prod_le_1
tff(fact_9055_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X2)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A2)),one_one_int) ) ).

% prod_le_1
tff(fact_9056_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X2)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),one_one_int) ) ).

% prod_le_1
tff(fact_9057_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_9058_bset_I1_J,axiom,
    ! [D4: int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_9059_bset_I2_J,axiom,
    ! [D4: int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_9060_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_9061_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_9062_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_la(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nat_diff_reindex
tff(fact_9063_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pm(G),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nat_diff_reindex
tff(fact_9064_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Na: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nb(G),Na),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) ).

% prod.atLeastAtMost_rev
tff(fact_9065_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Na: nat,M: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_pn(G),Na),M),aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M)) ).

% prod.atLeastAtMost_rev
tff(fact_9066_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = 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),Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9067_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9068_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = 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),Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9069_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_9070_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9071_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9072_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9073_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) = 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)),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_9074_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,Na))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9075_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,Na))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9076_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Na))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9077_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Na))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_9078_bset_I9_J,axiom,
    ! [D: int,D4: int,B2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb)) ) ) ) ).

% bset(9)
tff(fact_9079_bset_I10_J,axiom,
    ! [D: int,D4: int,B2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb)) ) ) ) ).

% bset(10)
tff(fact_9080_aset_I9_J,axiom,
    ! [D: int,D4: int,A2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb)) ) ) ) ).

% aset(9)
tff(fact_9081_aset_I10_J,axiom,
    ! [D: int,D4: int,A2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb)) ) ) ) ).

% aset(10)
tff(fact_9082_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_9083_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_9084_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = 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_da(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_9085_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_9086_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9087_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9088_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( 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),Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9089_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_9090_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_9091_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_9092_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = 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_da(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_9093_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_9094_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_da(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.atLeast1_atMost_eq
tff(fact_9095_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = groups705719431365010083at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.atLeast1_atMost_eq
tff(fact_9096_fact__prod,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))) ).

% fact_prod
tff(fact_9097_fact__prod,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,Na) = aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))) ).

% fact_prod
tff(fact_9098_fact__prod,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))) ).

% fact_prod
tff(fact_9099_fact__prod,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))) ).

% fact_prod
tff(fact_9100_fact__prod,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na))) ).

% fact_prod
tff(fact_9101_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_po(A)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pp(A),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nested_swap'
tff(fact_9102_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,Na: nat] : groups705719431365010083at_int(aTP_Lamm_pq(A),aa_nat_set_nat(set_ord_atMost_nat,Na)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ps(A),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nested_swap'
tff(fact_9103_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups6464643781859351333omplex(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_pt(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_9104_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups129246275422532515t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_pu(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_9105_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_pv(F),A,B,one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_9106_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_pw(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_9107_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_px(F),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_9108_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),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),Na))),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,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_9109_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),P3))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_9110_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),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),Na))),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,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_9111_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_9112_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K3: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X4: int] :
              ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_9113_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ).

% aset(7)
tff(fact_9114_aset_I5_J,axiom,
    ! [D4: int,Tb: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Tb,A2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb) ) ) ) ) ).

% aset(5)
tff(fact_9115_aset_I4_J,axiom,
    ! [D4: int,Tb: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Tb,A2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X3 != Tb )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4) != Tb ) ) ) ) ) ).

% aset(4)
tff(fact_9116_aset_I3_J,axiom,
    ! [D4: int,Tb: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Tb),one_one_int),A2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X3 = Tb )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4) = Tb ) ) ) ) ) ).

% aset(3)
tff(fact_9117_bset_I7_J,axiom,
    ! [D4: int,Tb: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Tb,B2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ).

% bset(7)
tff(fact_9118_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb) ) ) ) ).

% bset(5)
tff(fact_9119_bset_I4_J,axiom,
    ! [D4: int,Tb: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Tb,B2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X3 != Tb )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4) != Tb ) ) ) ) ) ).

% bset(4)
tff(fact_9120_bset_I3_J,axiom,
    ! [D4: int,Tb: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Tb),one_one_int),B2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X3 = Tb )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4) = Tb ) ) ) ) ) ).

% bset(3)
tff(fact_9121_norm__prod__diff,axiom,
    ! [I5: set_complex,Z: fun_complex_real,W: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_complex_real(Z,I2))),one_one_real) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_complex_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups766887009212190081x_real(Z,I5)),groups766887009212190081x_real(W,I5)))),groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_py(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9122_norm__prod__diff,axiom,
    ! [I5: set_real,Z: fun_real_real,W: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_pz(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9123_norm__prod__diff,axiom,
    ! [I5: set_set_nat,Z: fun_set_nat_real,W: fun_set_nat_real] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qa(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9124_norm__prod__diff,axiom,
    ! [I5: set_int,Z: fun_int_real,W: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( member_int(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qb(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9125_norm__prod__diff,axiom,
    ! [I5: set_complex,Z: fun_complex_complex,W: fun_complex_complex] :
      ( ! [I2: complex] :
          ( member_complex(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Z,I2))),one_one_real) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups3708469109370488835omplex(Z,I5)),groups3708469109370488835omplex(W,I5)))),groups5808333547571424918x_real(aa_fun1953249243402654830x_real(aTP_Lamm_qc(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9126_norm__prod__diff,axiom,
    ! [I5: set_real,Z: fun_real_complex,W: fun_real_complex] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qd(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9127_norm__prod__diff,axiom,
    ! [I5: set_set_nat,Z: fun_set_nat_complex,W: fun_set_nat_complex] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qe(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9128_norm__prod__diff,axiom,
    ! [I5: set_int,Z: fun_int_complex,W: fun_int_complex] :
      ( ! [I2: int] :
          ( member_int(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( member_int(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qf(Z),W),I5)) ) ) ).

% norm_prod_diff
tff(fact_9129_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: fun_nat_real,W: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qg(Z),W)),I5)) ) ) ).

% norm_prod_diff
tff(fact_9130_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: fun_nat_complex,W: fun_nat_complex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(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_qh(Z),W)),I5)) ) ) ).

% norm_prod_diff
tff(fact_9131_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_9132_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_fq(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_9133_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = 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_da(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_9134_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_db(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_9135_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Na)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_9136_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb) ) ) ) ).

% bset(6)
tff(fact_9137_bset_I8_J,axiom,
    ! [D4: int,Tb: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Tb),one_one_int),B2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ).

% bset(8)
tff(fact_9138_aset_I6_J,axiom,
    ! [D4: int,Tb: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Tb),one_one_int),A2)
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb) ) ) ) ) ).

% aset(6)
tff(fact_9139_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ).

% aset(8)
tff(fact_9140_cpmi,axiom,
    ! [D4: int,P: fun_int_o,P2: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z3)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P2,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa2: int] :
                  ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
         => ( ! [X2: int,K3: int] :
                ( aa_int_o(P2,X2)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D4))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P2,X4) )
                | ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,B2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_9141_cppi,axiom,
    ! [D4: int,P: fun_int_o,P2: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P2,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa2: int] :
                  ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
         => ( ! [X2: int,K3: int] :
                ( aa_int_o(P2,X2)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D4))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P2,X4) )
                | ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_9142_pochhammer__Suc__prod,axiom,
    ! [A: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Na)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qi(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod
tff(fact_9143_pochhammer__Suc__prod,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Na)) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_qj(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod
tff(fact_9144_pochhammer__Suc__prod,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Na)) = groups73079841787564623at_rat(aTP_Lamm_qk(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod
tff(fact_9145_pochhammer__Suc__prod,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod
tff(fact_9146_pochhammer__Suc__prod,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Na)) = groups705719431365010083at_int(aTP_Lamm_qm(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod
tff(fact_9147_pochhammer__prod__rev,axiom,
    ! [A: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),Na) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_qn(A),Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na)) ).

% pochhammer_prod_rev
tff(fact_9148_pochhammer__prod__rev,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),Na) = aa_set_nat_complex(groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_qo(A),Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na)) ).

% pochhammer_prod_rev
tff(fact_9149_pochhammer__prod__rev,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Na) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_qp(A),Na),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na)) ).

% pochhammer_prod_rev
tff(fact_9150_pochhammer__prod__rev,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Na) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qq(A),Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na)) ).

% pochhammer_prod_rev
tff(fact_9151_pochhammer__prod__rev,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Na) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qr(A),Na),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na)) ).

% pochhammer_prod_rev
tff(fact_9152_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_9153_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,Na: 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))),Na)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qs(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.in_pairs
tff(fact_9154_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,Na: 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))),Na)))) = groups73079841787564623at_rat(aTP_Lamm_qt(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.in_pairs
tff(fact_9155_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,Na: 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))),Na)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qu(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.in_pairs
tff(fact_9156_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,Na: 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))),Na)))) = groups705719431365010083at_int(aTP_Lamm_qv(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ).

% prod.in_pairs
tff(fact_9157_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Na: 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))),Na)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_qs(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_9158_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Na: 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))),Na)))) = groups73079841787564623at_rat(aTP_Lamm_qt(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_9159_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Na: 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))),Na)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qu(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_9160_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Na: 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))),Na)))) = groups705719431365010083at_int(aTP_Lamm_qv(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_9161_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Na: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,Na)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_qn(A),Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_9162_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Na)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_qo(A),Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_9163_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Na)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_qp(A),Na),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_9164_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qq(A),Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_9165_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Na)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qr(A),Na),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_9166_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_qw(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_hq(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_9167_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_qx(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_hy(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_9168_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_qy(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_hs(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_9169_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_qz(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_hw(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_9170_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ra(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_hu(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_9171_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(divide1717551699836669952omplex(aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_rb(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_9172_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_rc(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_9173_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_rd(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_9174_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_re,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_9175_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,groups705719431365010083at_int(aTP_Lamm_rf(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_9176_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_rg(L)),Qr) ).

% divmod_step_nat_def
tff(fact_9177_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_rh(L),Qr) ).

% divmod_step_int_def
tff(fact_9178_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_ri) ).

% pi_half
tff(fact_9179_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_ri)) ).

% pi_def
tff(fact_9180_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Na)
     => ( groups4538972089207619220nt_int(aTP_Lamm_an,set_or1266510415728281911st_int(M,Na)) = aa_int_int(aa_int_fun_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,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),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_9181_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_rg(L)),Qr) ).

% divmod_step_def
tff(fact_9182_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_rh(L),Qr) ).

% divmod_step_def
tff(fact_9183_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_rj(L),Qr) ).

% divmod_step_def
tff(fact_9184_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_rk,Y)) ).

% arcsin_def
tff(fact_9185_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_rl),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_9186_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_9187_valid__eq,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Tb),D)
    <=> aa_nat_o(vEBT_invar_vebt(Tb),D) ) ).

% valid_eq
tff(fact_9188_valid__eq1,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tb),D)
     => aa_nat_o(vEBT_VEBT_valid(Tb),D) ) ).

% valid_eq1
tff(fact_9189_valid__eq2,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Tb),D)
     => aa_nat_o(vEBT_invar_vebt(Tb),D) ) ).

% valid_eq2
tff(fact_9190_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_fc(F),A2) ).

% int_prod
tff(fact_9191_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_fd(F),A2) ).

% int_prod
tff(fact_9192_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9193_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_an,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_9194_divmod__nat__def,axiom,
    ! [M: nat,Na: nat] : divmod_nat(M,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ).

% divmod_nat_def
tff(fact_9195_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_9196_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = 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,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_9197_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_9198_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Na)
          & aa_nat_o(P,M5) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_9199_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Na)
         => aa_nat_o(P,M5) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less_eq
tff(fact_9200_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9201_lessThan__atLeast0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Na) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na) ).

% lessThan_atLeast0
tff(fact_9202_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_9203_prod__Suc__fact,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_fact
tff(fact_9204_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rm(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)))),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),Na))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9205_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_9206_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9207_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9208_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X6,M5)),aa_nat_real(X6,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9209_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9210_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_rj(L),Qr) ).

% divmod_step_integer_def
tff(fact_9211_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9212_Re__divide__of__nat,axiom,
    ! [Z: complex,Na: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,Na))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% Re_divide_of_nat
tff(fact_9213_Re__divide__of__real,axiom,
    ! [Z: complex,R4: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R4))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),R4) ).

% Re_divide_of_real
tff(fact_9214_Re__sgn,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Re_sgn
tff(fact_9215_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),numeral_numeral_real(W)) ).

% Re_divide_numeral
tff(fact_9216_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9217_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9218_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(M,Na) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Na)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Na))) ).

% divmod_integer'_def
tff(fact_9219_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9220_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9221_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y)) ).

% plus_complex.simps(1)
tff(fact_9222_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9223_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9224_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_9225_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9226_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(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_9227_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9228_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,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9229_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9230_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(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9231_Im__divide__of__real,axiom,
    ! [Z: complex,R4: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R4))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),R4) ).

% Im_divide_of_real
tff(fact_9232_Im__sgn,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Im_sgn
tff(fact_9233_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),numeral_numeral_real(W)) ).

% Im_divide_numeral
tff(fact_9234_Im__divide__of__nat,axiom,
    ! [Z: complex,Na: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,Na))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% Im_divide_of_nat
tff(fact_9235_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,X)),zero_zero_real)
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(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_9236_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9237_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),X)) ).

% divide_integer.abs_eq
tff(fact_9238_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_9239_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs_Code_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).

% abs_integer_code
tff(fact_9240_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X) ) ).

% less_integer.abs_eq
tff(fact_9241_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9242_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9243_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),X)) ).

% plus_integer.abs_eq
tff(fact_9244_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_9245_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X) ) ).

% less_eq_integer.abs_eq
tff(fact_9246_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y)) ).

% plus_complex.simps(2)
tff(fact_9247_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_9248_plus__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(plus_plus_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% plus_complex.code
tff(fact_9249_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(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 )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_9250_cmod__le,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(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_9251_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_9252_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_9253_cmod__power2,axiom,
    ! [Z: complex] : aa_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(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9254_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,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).

% Im_power2
tff(fact_9255_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(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9256_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9257_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(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9258_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(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9259_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9260_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(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9261_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 )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9262_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(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_9263_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(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9264_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(power_power_real(aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9265_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_9266_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_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(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_9267_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(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9268_Im__Reals__divide,axiom,
    ! [R4: complex,Z: complex] :
      ( member_complex(R4,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R4),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,R4))),aa_complex_real(im,Z))),aa_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_9269_Re__Reals__divide,axiom,
    ! [R4: complex,Z: complex] :
      ( member_complex(R4,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R4),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R4)),aa_complex_real(re,Z))),aa_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_9270_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(power_power_real(aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9271_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_divide
tff(fact_9272_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_9273_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_9274_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_add
tff(fact_9275_Re__divide__Reals,axiom,
    ! [R4: complex,Z: complex] :
      ( member_complex(R4,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),R4)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(re,R4)) ) ) ).

% Re_divide_Reals
tff(fact_9276_Im__divide__Reals,axiom,
    ! [R4: complex,Z: complex] :
      ( member_complex(R4,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),R4)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(re,R4)) ) ) ).

% Im_divide_Reals
tff(fact_9277_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_9278_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_9279_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(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_9280_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(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_9281_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(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_9282_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(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_9283_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(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_9284_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(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_9285_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(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_9286_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(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_9287_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(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_9288_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(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_9289_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_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_9290_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9291_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9292_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(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_9293_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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_9294_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      code_integer_of_num(aa_num_num(bit1,Na)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Na),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9295_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_9296_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).

% divmod_integer_def
tff(fact_9297_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_9298_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9299_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      code_integer_of_num(aa_num_num(bit0,Na)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Na),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9300_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_9301_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_rn(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9302_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_ro(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_rp(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9303_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_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9304_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9305_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9306_or__minus__numerals_I6_J,axiom,
    ! [Na: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))) ).

% or_minus_numerals(6)
tff(fact_9307_or__minus__numerals_I2_J,axiom,
    ! [Na: 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,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))) ).

% or_minus_numerals(2)
tff(fact_9308_and__minus__minus__numerals,axiom,
    ! [M: num,Na: 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,Na))) = 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,Na)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9309_or__minus__minus__numerals,axiom,
    ! [M: num,Na: 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,Na))) = 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,Na)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9310_bit__or__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se1409905431419307370or_int(K),L)),Na)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Na)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Na) ) ) ).

% bit_or_int_iff
tff(fact_9311_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(X),Y)) ) ) ).

% OR_lower
tff(fact_9312_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L)) ) ).

% or_greater_eq
tff(fact_9313_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_9314_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_rq),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9315_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_rr),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9316_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_9317_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_9318_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_9319_concat__bit__def,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Na,K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(Na),K)),aa_int_int(bit_se545348938243370406it_int(Na),L)) ).

% concat_bit_def
tff(fact_9320_set__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K) = aa_int_int(bit_se1409905431419307370or_int(K),aa_int_int(bit_se545348938243370406it_int(Na),one_one_int)) ).

% set_bit_int_def
tff(fact_9321_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_9322_or__not__numerals_I2_J,axiom,
    ! [Na: 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,Na)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) ).

% or_not_numerals(2)
tff(fact_9323_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_9324_diff__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_am),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ).

% diff_Suc
tff(fact_9325_or__not__numerals_I3_J,axiom,
    ! [Na: 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,Na)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))) ).

% or_not_numerals(3)
tff(fact_9326_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_9327_or__not__numerals_I6_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na)))) ).

% or_not_numerals(6)
tff(fact_9328_OR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(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))),Na)) ) ) ) ).

% OR_upper
tff(fact_9329_or__not__numerals_I5_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na))))) ).

% or_not_numerals(5)
tff(fact_9330_or__not__numerals_I9_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na))))) ).

% or_not_numerals(9)
tff(fact_9331_or__not__numerals_I8_J,axiom,
    ! [M: num,Na: 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,Na)))) = 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,Na))))) ).

% or_not_numerals(8)
tff(fact_9332_or__minus__numerals_I1_J,axiom,
    ! [Na: 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,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(one,bitM(Na)))) ).

% or_minus_numerals(1)
tff(fact_9333_or__minus__numerals_I5_J,axiom,
    ! [Na: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(one,bitM(Na)))) ).

% or_minus_numerals(5)
tff(fact_9334_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9335_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_9336_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_9337_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_9338_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_9339_or__minus__numerals_I8_J,axiom,
    ! [Na: 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,Na)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,aa_num_num(bit0,Na)))) ).

% or_minus_numerals(8)
tff(fact_9340_or__minus__numerals_I4_J,axiom,
    ! [M: num,Na: 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,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,aa_num_num(bit0,Na)))) ).

% or_minus_numerals(4)
tff(fact_9341_or__minus__numerals_I3_J,axiom,
    ! [M: num,Na: 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,Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(3)
tff(fact_9342_or__minus__numerals_I7_J,axiom,
    ! [Na: 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,Na)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(7)
tff(fact_9343_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one,one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9344_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9345_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(aa_num_num(bit0,Na),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9346_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa_num_num(bit0,Na),aa_num_num(bit1,M)) = aa_num_num(bit0,bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(6)
tff(fact_9347_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(aa_num_num(bit1,Na),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9348_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: 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_9349_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa_num_num(bit0,Na),aa_num_num(bit0,M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(5)
tff(fact_9350_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa_num_num(bit1,Na),aa_num_num(bit1,M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(9)
tff(fact_9351_or__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% or_nat_def
tff(fact_9352_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: 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_9353_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa_num_num(bit1,Na),aa_num_num(bit0,M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(8)
tff(fact_9354_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y: 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) ) ) )
           => ( ( ? [N: num] : X = aa_num_num(bit0,N)
               => ( ( Xa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( X = aa_num_num(bit0,N) )
                   => ! [M4: num] :
                        ( ( Xa = aa_num_num(bit0,M4) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
               => ( ! [N: num] :
                      ( ( X = aa_num_num(bit0,N) )
                     => ! [M4: num] :
                          ( ( Xa = aa_num_num(bit1,M4) )
                         => ( Y != aa_num_num(bit0,bit_or_not_num_neg(N,M4)) ) ) )
                 => ( ( ? [N: num] : X = aa_num_num(bit1,N)
                     => ( ( Xa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( X = aa_num_num(bit1,N) )
                         => ! [M4: num] :
                              ( ( Xa = aa_num_num(bit0,M4) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X = aa_num_num(bit1,N) )
                           => ! [M4: num] :
                                ( ( Xa = aa_num_num(bit1,M4) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9355_numeral__or__not__num__eq,axiom,
    ! [M: num,Na: num] : aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,Na)) = 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,Na)))) ).

% numeral_or_not_num_eq
tff(fact_9356_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Na: 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,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(Na,M))) ).

% int_numeral_not_or_num_neg
tff(fact_9357_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Na: 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,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,Na))) ).

% int_numeral_or_not_num_neg
tff(fact_9358_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aTP_Lamm_rs(X)) ).

% floor_real_def
tff(fact_9359_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% Suc_0_or_eq
tff(fact_9360_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% or_Suc_0_eq
tff(fact_9361_or__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ))),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9362_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_9363_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_9364_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)) ).

% max_Suc_Suc
tff(fact_9365_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_9366_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_9367_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_9368_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_9369_max__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ).

% max_0L
tff(fact_9370_max__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ).

% max_0R
tff(fact_9371_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% max_numeral_Suc
tff(fact_9372_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9373_nat__mult__max__right,axiom,
    ! [M: nat,Na: 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,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_max_right
tff(fact_9374_nat__mult__max__left,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),Q3)) ).

% nat_mult_max_left
tff(fact_9375_nat__add__max__left,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),Q3)) ).

% nat_add_max_left
tff(fact_9376_nat__add__max__right,axiom,
    ! [M: nat,Na: 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,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ).

% nat_add_max_right
tff(fact_9377_nat__minus__add__max,axiom,
    ! [Na: 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,Na),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),M) ).

% nat_minus_add_max
tff(fact_9378_max__Suc1,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_rt,Na)),M) ).

% max_Suc1
tff(fact_9379_max__Suc2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_ru,Na)),M) ).

% max_Suc2
tff(fact_9380_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,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,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9381_floor__rat__def,axiom,
    ! [X: rat] : archim3151403230148437115or_rat(X) = the_int(aTP_Lamm_rv(X)) ).

% floor_rat_def
tff(fact_9382_abs__rat__def,axiom,
    ! [A: rat] :
      abs_abs_rat(A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),uminus_uminus_rat(A),A) ).

% abs_rat_def
tff(fact_9383_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9384_obtain__pos__sum,axiom,
    ! [R4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R4)
     => ~ ! [S3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S3)
           => ! [T: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T)
               => ( R4 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S3),T) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9385_sgn__rat__def,axiom,
    ! [A: rat] :
      sgn_sgn_rat(A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9386_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_rw,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9387_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9388_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% quotient_of_number(3)
tff(fact_9389_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_9390_rat__one__code,axiom,
    quotient_of(one_one_rat) = product_Pair_int_int(one_one_int,one_one_int) ).

% rat_one_code
tff(fact_9391_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_9392_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(5)
tff(fact_9393_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int),one_one_int) ).

% quotient_of_number(4)
tff(fact_9394_divide__rat__def,axiom,
    ! [Q3: rat,R4: rat] : divide_divide_rat(Q3,R4) = aa_rat_rat(times_times_rat(Q3),aa_rat_rat(inverse_inverse_rat,R4)) ).

% divide_rat_def
tff(fact_9395_diff__rat__def,axiom,
    ! [Q3: rat,R4: rat] : minus_minus_rat(Q3,R4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q3),uminus_uminus_rat(R4)) ).

% diff_rat_def
tff(fact_9396_rat__divide__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(divide_divide_rat(P3,Q3)) = produc4245557441103728435nt_int(aTP_Lamm_ry(Q3),quotient_of(P3)) ).

% rat_divide_code
tff(fact_9397_quotient__of__div,axiom,
    ! [R4: rat,Na: int,D: int] :
      ( ( quotient_of(R4) = product_Pair_int_int(Na,D) )
     => ( R4 = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Na),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9398_rat__plus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P3),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_sa(Q3),quotient_of(P3)) ).

% rat_plus_code
tff(fact_9399_quotient__of__denom__pos,axiom,
    ! [R4: rat,P3: int,Q3: int] :
      ( ( quotient_of(R4) = product_Pair_int_int(P3,Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% quotient_of_denom_pos
tff(fact_9400_normalize__denom__pos,axiom,
    ! [R4: product_prod_int_int,P3: int,Q3: int] :
      ( ( normalize(R4) = product_Pair_int_int(P3,Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% normalize_denom_pos
tff(fact_9401_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P3: int,R4: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P3,Q3)) = normalize(product_Pair_int_int(R4,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,R4),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9402_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_am),Nat) ).

% pred_def
tff(fact_9403_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = product_Pair_int_int(A,one_one_int) ).

% quotient_of_int
tff(fact_9404_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9405_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),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_9406_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)),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_9407_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_9408_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_9409_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_9410_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,Na: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Na) )
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_9411_finite__nat__set__iff__bounded,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_9412_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sb(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_9413_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sc(F),U))) ) ).

% finite_less_ub
tff(fact_9414_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_9415_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,M))) ) ).

% finite_divisors_nat
tff(fact_9416_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,Na: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_9417_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,Na: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9418_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_9419_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_9420_Frct__code__post_I3_J,axiom,
    frct(product_Pair_int_int(one_one_int,one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9421_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_9422_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_9423_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fz,K))) ).

% finite_Collect_le_nat
tff(fact_9424_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jo,K))) ).

% finite_Collect_less_nat
tff(fact_9425_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_sd(A),B))) ).

% finite_interval_int1
tff(fact_9426_finite__interval__int4,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_se(A),B))) ).

% finite_interval_int4
tff(fact_9427_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_sf(A),B))) ).

% finite_interval_int2
tff(fact_9428_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_sg(A),B))) ).

% finite_interval_int3
tff(fact_9429_finite__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_kn(Na),C))) ) ).

% finite_nth_roots
tff(fact_9430_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_finite_int(set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9431_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ag,I))) ) ).

% finite_divisors_int
tff(fact_9432_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => 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,Na),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,Na))))),collect_complex(aTP_Lamm_ko(Na)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_sh(C),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9433_set__encode__insert,axiom,
    ! [A2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Na),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Na,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_9434_real__root__zero,axiom,
    ! [Na: nat] : aa_real_real(aa_nat_fun_real_real(root,Na),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9435_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_9436_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_9437_real__root__eq__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = aa_real_real(aa_nat_fun_real_real(root,Na),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9438_real__root__eq__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9439_real__root__less__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ).

% real_root_less_iff
tff(fact_9440_real__root__le__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% real_root_le_iff
tff(fact_9441_real__root__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9442_real__root__eq__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9443_real__root__lt__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9444_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9445_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9446_real__root__le__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9447_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9448_real__root__lt__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9449_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9450_real__root__le__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9451_real__root__pow__pos2,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_9452_real__root__minus,axiom,
    ! [Na: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)) ).

% real_root_minus
tff(fact_9453_real__root__mult__exp,axiom,
    ! [M: nat,Na: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),X) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ).

% real_root_mult_exp
tff(fact_9454_real__root__mult,axiom,
    ! [Na: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Na),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,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ).

% real_root_mult
tff(fact_9455_real__root__divide,axiom,
    ! [Na: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Na),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,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ).

% real_root_divide
tff(fact_9456_real__root__commute,axiom,
    ! [M: nat,Na: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Na),X)) = aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(aa_nat_fun_real_real(root,M),X)) ).

% real_root_commute
tff(fact_9457_real__root__inverse,axiom,
    ! [Na: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)) ).

% real_root_inverse
tff(fact_9458_real__root__pos__pos__le,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ).

% real_root_pos_pos_le
tff(fact_9459_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_9460_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_9461_real__root__less__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_9462_real__root__le__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_9463_real__root__power,axiom,
    ! [Na: nat,X: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(power_power_real(X),K)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),K) ) ) ).

% real_root_power
tff(fact_9464_real__root__abs,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_abs
tff(fact_9465_sgn__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_9466_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_9467_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(Na,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_9468_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_9469_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9470_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9471_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_9472_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_9473_real__root__gt__zero,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_gt_zero
tff(fact_9474_real__root__strict__decreasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9475_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_9476_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(power_power_real(Y),Na))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9477_real__root__pos__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_pos_pos
tff(fact_9478_real__root__strict__increasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,N2),X)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9479_real__root__decreasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ) ).

% real_root_decreasing
tff(fact_9480_real__root__pow__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos
tff(fact_9481_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(power_power_real(Y),Na) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9482_real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(power_power_real(X),Na)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_9483_odd__real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(power_power_real(X),Na)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_9484_odd__real__root__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( aa_nat_real(power_power_real(Y),Na) = X )
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9485_odd__real__root__pow,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ).

% odd_real_root_pow
tff(fact_9486_real__root__increasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,N2),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_9487_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_9488_sgn__power__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),X))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),X))),Na)) = X ) ) ).

% sgn_power_root
tff(fact_9489_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Na),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))),Na)),Z)) = insert_nat(Na,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9490_ln__root,axiom,
    ! [Na: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Na),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_9491_log__root,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Na),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_9492_log__base__root,axiom,
    ! [Na: nat,B: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Na),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% log_base_root
tff(fact_9493_split__root,axiom,
    ! [P: fun_real_o,Na: nat,X: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Na),X))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ! [Y4: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y4)),Na)) = X )
             => aa_real_o(P,Y4) ) ) ) ) ).

% split_root
tff(fact_9494_root__powr__inverse,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),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,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_9495_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ finite_finite_int(S2)
    <=> ! [M5: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M5),aa_int_int(abs_abs_int,N4))
          & member_int(N4,S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_9496_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ finite_finite_int(S2)
    <=> ! [M5: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N4))
          & member_int(N4,S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_9497_card__Collect__less__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jo,Na))) = Na ).

% card_Collect_less_nat
tff(fact_9498_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9499_card__Collect__le__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fz,Na))) = aa_nat_nat(suc,Na) ).

% card_Collect_le_nat
tff(fact_9500_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9501_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_9502_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_si(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9503_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sj(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_si(M7),I))) ) ) ).

% card_less_Suc
tff(fact_9504_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sj(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_si(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_9505_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_9506_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9507_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na),set_or1266510415728281911st_int(M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9508_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,Na: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N2)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9509_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),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_am),S2)) ).

% card_sum_le_nat_sum
tff(fact_9510_card__nth__roots,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_sh(C),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_9511_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( finite_card_complex(collect_complex(aTP_Lamm_ko(Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_9512_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9513_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M4)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N7)
              & aa_set_nat_o(member_nat(N7),S2) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S2) ) ).

% unbounded_k_infinite
tff(fact_9514_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_9515_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa) = Y )
     => ( Y = $ite(
            ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                  & ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9516_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & 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(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_9517_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,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_9518_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K,L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
            ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & 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(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_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(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9519_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))
       => ~ ( ( Y = $ite(
                  ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                        & ~ aa_int_o(aa_int_fun_int_o(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(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_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_9520_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_9521_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9522_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_9523_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9524_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_9525_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_9526_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_9527_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9528_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_9529_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),insert_nat(Na,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9530_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Na),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9531_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Na),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9532_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9533_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K3: int,L2: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K3,L2))
           => ( ( ~ ( member_int(K3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K3),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9534_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I2: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I2,J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9535_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_9536_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9537_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_9538_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9539_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9540_drop__bit__minus__one,axiom,
    ! [Na: nat] : aa_int_int(bit_se8568078237143864401it_int(Na),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9541_fst__divmod__nat,axiom,
    ! [M: nat,Na: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) ).

% fst_divmod_nat
tff(fact_9542_snd__divmod__nat,axiom,
    ! [M: nat,Na: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) ).

% snd_divmod_nat
tff(fact_9543_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9544_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_9545_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9546_drop__bit__push__bit__int,axiom,
    ! [M: nat,Na: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),aa_int_int(bit_se545348938243370406it_int(Na),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9547_drop__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Na),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% drop_bit_int_def
tff(fact_9548_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [R2: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R2),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2)))
          & ! [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N7),aa_set_nat_nat(finite_card_nat,S2))
             => aa_set_nat_o(member_nat(aa_nat_nat(R2,N7)),S2) ) ) ) ).

% finite_enumerate
tff(fact_9549_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_integer(K,L)) = divide6298287555418463151nteger(K,L) ).

% fst_divmod_integer
tff(fact_9550_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_abs(K,L)) = divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).

% fst_divmod_abs
tff(fact_9551_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9552_drop__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(Na),K)) ).

% drop_bit_nat_eq
tff(fact_9553_quotient__of__denom__pos_H,axiom,
    ! [R4: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R4))) ).

% quotient_of_denom_pos'
tff(fact_9554_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_9555_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),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_9556_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( Y = $ite(Xa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),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_9557_drop__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),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))),Na)) ).

% drop_bit_nat_def
tff(fact_9558_rat__sgn__code,axiom,
    ! [P3: rat] : quotient_of(sgn_sgn_rat(P3)) = product_Pair_int_int(aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P3))),one_one_int) ).

% rat_sgn_code
tff(fact_9559_nat__descend__induct,axiom,
    ! [Na: nat,P: fun_nat_o,M: nat] :
      ( ! [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K3)
         => aa_nat_o(P,K3) )
     => ( ! [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Na)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K3) ) )
       => aa_nat_o(P,M) ) ) ).

% nat_descend_induct
tff(fact_9560_one__mod__minus__numeral,axiom,
    ! [Na: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(one,Na)))) ).

% one_mod_minus_numeral
tff(fact_9561_minus__one__mod__numeral,axiom,
    ! [Na: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(one,Na))) ).

% minus_one_mod_numeral
tff(fact_9562_minus__numeral__mod__numeral,axiom,
    ! [M: num,Na: 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,Na)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(M,Na))) ).

% minus_numeral_mod_numeral
tff(fact_9563_numeral__mod__minus__numeral,axiom,
    ! [M: num,Na: 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,Na))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(M,Na)))) ).

% numeral_mod_minus_numeral
tff(fact_9564_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R4: int] :
      aa_int_int(adjust_mod(L),R4) = $ite(R4 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R4)) ).

% Divides.adjust_mod_def
tff(fact_9565_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))
       => ~ ( ( Y = $ite(Xa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),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_9566_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      normalize(P3) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P3)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)),
          product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a)) ),
        $ite(
          product_snd_int_int(P3) = zero_zero_int,
          product_Pair_int_int(zero_zero_int,one_one_int),
          $let(
            a: int,
            a:= 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))),
            product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a)) ) ) ) ).

% normalize_def
tff(fact_9567_gcd__1__int,axiom,
    ! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9568_gcd__pos__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),Na))
    <=> ( ( M != zero_zero_int )
        | ( Na != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9569_gcd__neg__numeral__1__int,axiom,
    ! [Na: 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,Na))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,Na)),X) ).

% gcd_neg_numeral_1_int
tff(fact_9570_gcd__neg__numeral__2__int,axiom,
    ! [X: int,Na: 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,Na))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,Na)) ).

% gcd_neg_numeral_2_int
tff(fact_9571_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_9572_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_9573_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(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_9574_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% bezout_int
tff(fact_9575_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ) ).

% gcd_le1_int
tff(fact_9576_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ) ).

% gcd_le2_int
tff(fact_9577_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(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))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9578_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9579_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9580_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,
        $ite(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_9581_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),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_9582_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_sk(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9583_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_9584_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_9585_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_9586_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_9587_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_9588_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_9589_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9590_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_9591_gcd__pos__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na))
    <=> ( ( M != zero_zero_nat )
        | ( Na != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9592_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9593_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9594_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) ) ) ).

% gcd_diff1_nat
tff(fact_9595_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) ) ) ).

% gcd_diff2_nat
tff(fact_9596_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y )
     => ( Y = $ite(Xa = zero_zero_nat,X,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_9597_gcd__nat_Osimps,axiom,
    ! [X: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = $ite(Y = zero_zero_nat,X,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_9598_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_9599_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X2: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9600_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X2: nat,Y2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9601_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_9602_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))
       => ~ ( ( Y = $ite(Xa = zero_zero_nat,X,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_9603_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_9604_card_Ocomp__fun__commute__on,axiom,
    comp_nat_nat_nat(suc,suc) = comp_nat_nat_nat(suc,suc) ).

% card.comp_fun_commute_on
tff(fact_9605_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9606_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_9607_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Na: num] : aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K),neg_numeral_sub_int(Na,one))) ).

% xor_minus_numerals(2)
tff(fact_9608_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9609_xor__minus__numerals_I1_J,axiom,
    ! [Na: num,K: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(neg_numeral_sub_int(Na,one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9610_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9611_tanh__real__bounds,axiom,
    ! [X: real] : member_real(aa_real_real(tanh_real,X),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9612_sub__BitM__One__eq,axiom,
    ! [Na: num] : neg_numeral_sub_int(bitM(Na),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),neg_numeral_sub_int(Na,one)) ).

% sub_BitM_One_eq
tff(fact_9613_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9614_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_fz,aTP_Lamm_jo) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9615_Suc__funpow,axiom,
    ! [Na: nat] : compow_fun_nat_nat(Na,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Na) ).

% Suc_funpow
tff(fact_9616_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9617_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_9618_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9619_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_sl) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9620_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_sm,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9621_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_sn,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9622_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_sp),Xa),X)) ).

% times_int.abs_eq
tff(fact_9623_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9624_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(numera6620942414471956472nteger(K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9625_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(X,Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% plus_integer.rep_eq
tff(fact_9626_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9627_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(divide6298287555418463151nteger(X,Xa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% divide_integer.rep_eq
tff(fact_9628_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) ).

% eq_Abs_Integ
tff(fact_9629_int_Oabs__induct,axiom,
    ! [P: fun_int_o,X: int] :
      ( ! [Y2: product_prod_nat_nat] : aa_int_o(P,aa_Pro4240683299535347697at_int(abs_Integ,Y2))
     => aa_int_o(P,X) ) ).

% int.abs_induct
tff(fact_9630_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ) ).

% less_integer.rep_eq
tff(fact_9631_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_9632_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9633_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_9634_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_9635_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_9636_int__def,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,Na) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Na),zero_zero_nat)) ).

% int_def
tff(fact_9637_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_sq),X)) ).

% uminus_int.abs_eq
tff(fact_9638_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_9639_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_ss),Xa),X) ) ).

% less_int.abs_eq
tff(fact_9640_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_su),Xa),X) ) ).

% less_eq_int.abs_eq
tff(fact_9641_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_sw),Xa),X)) ).

% plus_int.abs_eq
tff(fact_9642_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_sy),Xa),X)) ).

% minus_int.abs_eq
tff(fact_9643_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( aa_set_nat_o(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_9644_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na),inc(aa_nat_num(num_of_nat,Na)),one) ).

% num_of_nat.simps(2)
tff(fact_9645_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_9646_Gcd__nat__eq__one,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),N2)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N2) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9647_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9648_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_9649_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),one_one_nat)
     => ( aa_nat_num(num_of_nat,Na) = one ) ) ).

% num_of_nat_One
tff(fact_9650_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Na)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_9651_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_plus_num(aa_nat_num(num_of_nat,M),aa_nat_num(num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9652_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_su),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_eq_int.rep_eq
tff(fact_9653_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_ss),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_int.rep_eq
tff(fact_9654_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_9655_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_9656_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_sq)) ).

% uminus_int_def
tff(fact_9657_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_sz) ).

% prod_encode_def
tff(fact_9658_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(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_9659_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(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_9660_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_9661_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sp)) ).

% times_int_def
tff(fact_9662_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sy)) ).

% minus_int_def
tff(fact_9663_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sw)) ).

% plus_int_def
tff(fact_9664_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( 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))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9665_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xa )
                  & $let(
                      n2: nat,
                      n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( 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))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9666_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n2: nat,
            n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n2),
              ( ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X7)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Dega),TreeLista),Summarya),n2),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9667_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9668_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( ( ? [Uu2: $o,Uv2: $o] : X = vEBT_Leaf((Uu2),(Uv2))
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xa )
                & $let(
                    n2: nat,
                    n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( 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))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9669_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( 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))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9670_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa))
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xa))
                 => ~ ( ( Deg = Xa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( 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))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9671_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xa))
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xa))
                 => ( ( Deg = Xa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n2) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( 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))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Deg),TreeList),Summary),n2),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9672_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Na: 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,Na))) = case_option_int_num(zero_zero_int,aTP_Lamm_tb(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),Na)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9673_and__minus__numerals_I3_J,axiom,
    ! [M: num,Na: 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,Na)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(Na))) ).

% and_minus_numerals(3)
tff(fact_9674_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_9675_take__bit__num__simps_I2_J,axiom,
    ! [Na: nat] : bit_take_bit_num(aa_nat_nat(suc,Na),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9676_take__bit__num__simps_I5_J,axiom,
    ! [R4: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R4),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9677_take__bit__num__simps_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_tc,bit_take_bit_num(Na,M)) ).

% take_bit_num_simps(3)
tff(fact_9678_take__bit__num__simps_I4_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Na,M))) ).

% take_bit_num_simps(4)
tff(fact_9679_take__bit__num__simps_I6_J,axiom,
    ! [R4: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R4),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_tc,bit_take_bit_num(aa_num_nat(pred_numeral,R4),M)) ).

% take_bit_num_simps(6)
tff(fact_9680_take__bit__num__simps_I7_J,axiom,
    ! [R4: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R4),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R4),M))) ).

% take_bit_num_simps(7)
tff(fact_9681_and__minus__numerals_I8_J,axiom,
    ! [Na: 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,Na)))),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,Na))) ).

% and_minus_numerals(8)
tff(fact_9682_and__minus__numerals_I4_J,axiom,
    ! [M: num,Na: 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,Na)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,Na))) ).

% and_minus_numerals(4)
tff(fact_9683_and__minus__numerals_I7_J,axiom,
    ! [Na: 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,Na)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(Na))) ).

% and_minus_numerals(7)
tff(fact_9684_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9685_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_td(M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9686_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Na: nat] : bit_take_bit_num(Na,one) = case_nat_option_num(none_num,aTP_Lamm_te,Na) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9687_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_9688_and__not__num_Osimps_I2_J,axiom,
    ! [Na: num] : bit_and_not_num(one,aa_num_num(bit0,Na)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9689_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9690_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9691_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9692_and__not__num_Osimps_I3_J,axiom,
    ! [Na: num] : bit_and_not_num(one,aa_num_num(bit1,Na)) = none_num ).

% and_not_num.simps(3)
tff(fact_9693_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_tf(M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9694_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_9695_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Na: num,Q3: num] :
      ( ( bit_and_not_num(M,Na) = 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,Na))) = aa_num_int(numeral_numeral_int,Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9696_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tg,bit_and_not_num(M,Na)) ).

% and_not_num.simps(8)
tff(fact_9697_and__not__num__eq__None__iff,axiom,
    ! [M: num,Na: num] :
      ( ( bit_and_not_num(M,Na) = 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,Na))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9698_int__numeral__not__and__num,axiom,
    ! [M: num,Na: 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,Na)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Na,M)) ).

% int_numeral_not_and_num
tff(fact_9699_int__numeral__and__not__num,axiom,
    ! [M: num,Na: 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,Na))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,Na)) ).

% int_numeral_and_not_num
tff(fact_9700_take__bit__num__def,axiom,
    ! [Na: nat,M: num] :
      bit_take_bit_num(Na,M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_9701_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9702_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : pow(X,aa_num_num(bit1,Y)) = times_times_num(sqr(pow(X,Y)),X) ).

% pow.simps(3)
tff(fact_9703_sqr_Osimps_I2_J,axiom,
    ! [Na: num] : sqr(aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(bit0,sqr(Na))) ).

% sqr.simps(2)
tff(fact_9704_sqr_Osimps_I1_J,axiom,
    sqr(one) = one ).

% sqr.simps(1)
tff(fact_9705_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9706_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = times_times_num(X,X) ).

% sqr_conv_mult
tff(fact_9707_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : pow(X,aa_num_num(bit0,Y)) = sqr(pow(X,Y)) ).

% pow.simps(2)
tff(fact_9708_sqr_Osimps_I3_J,axiom,
    ! [Na: num] : sqr(aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(bit0,plus_plus_num(sqr(Na),Na))) ).

% sqr.simps(3)
tff(fact_9709_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 )
           => ( ? [N: num] : Xa = aa_num_num(bit0,N)
             => ( Y != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] : Xa = aa_num_num(bit1,N)
               => ( 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) )
                   => ! [N: num] :
                        ( ( Xa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M4,N)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M4,N)) ) ) )
                 => ( ! [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) )
                         => ! [N: num] :
                              ( ( Xa = aa_num_num(bit0,N) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tg,bit_and_not_num(M4,N)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M4,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9710_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,M) = produc478579273971653890on_num(aTP_Lamm_tk,product_Pair_nat_num(Na,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9711_xor__num_Osimps_I6_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),Na))) ).

% xor_num.simps(6)
tff(fact_9712_xor__num_Osimps_I5_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),Na)) ).

% xor_num.simps(5)
tff(fact_9713_xor__num_Osimps_I9_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),Na)) ).

% xor_num.simps(9)
tff(fact_9714_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit0,Na)) = map_option_num_num(bit0,bit_and_not_num(M,Na)) ).

% and_not_num.simps(5)
tff(fact_9715_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = map_option_num_num(bit0,bit_and_not_num(M,Na)) ).

% and_not_num.simps(6)
tff(fact_9716_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = map_option_num_num(bit0,bit_and_not_num(M,Na)) ).

% and_not_num.simps(9)
tff(fact_9717_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_9718_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9719_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 )
           => ! [N: num] :
                ( ( Xa = aa_num_num(bit0,N) )
               => ( Y != some_num(aa_num_num(bit1,N)) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa = aa_num_num(bit1,N) )
                 => ( Y != some_num(aa_num_num(bit0,N)) ) ) )
           => ( ! [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) )
                   => ! [N: num] :
                        ( ( Xa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit1,N) )
                         => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N))) ) ) )
                 => ( ! [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) )
                         => ! [N: num] :
                              ( ( Xa = aa_num_num(bit0,N) )
                             => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N))) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9720_xor__num_Osimps_I2_J,axiom,
    ! [Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,Na)) = some_num(aa_num_num(bit1,Na)) ).

% xor_num.simps(2)
tff(fact_9721_xor__num_Osimps_I3_J,axiom,
    ! [Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,Na)) = some_num(aa_num_num(bit0,Na)) ).

% xor_num.simps(3)
tff(fact_9722_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_9723_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_9724_xor__num_Osimps_I8_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),Na))) ).

% xor_num.simps(8)
tff(fact_9725_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 )
           => ( ? [N: num] : Xa = aa_num_num(bit0,N)
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] : Xa = aa_num_num(bit1,N)
               => ( 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) )
                   => ! [N: num] :
                        ( ( Xa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa_num_num(bit0,M4) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) ) ) )
                 => ( ( ? [M4: num] : X = aa_num_num(bit1,M4)
                     => ( ( Xa = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa_num_num(bit1,M4) )
                         => ! [N: num] :
                              ( ( Xa = aa_num_num(bit0,N) )
                             => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa_num_num(bit1,M4) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit1,N) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tg,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9726_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_9727_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_9728_and__num_Osimps_I5_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),Na)) ).

% and_num.simps(5)
tff(fact_9729_and__num_Osimps_I3_J,axiom,
    ! [Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,Na)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9730_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_9731_and__num_Osimps_I2_J,axiom,
    ! [Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,Na)) = none_num ).

% and_num.simps(2)
tff(fact_9732_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_9733_and__num_Osimps_I8_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),Na)) ).

% and_num.simps(8)
tff(fact_9734_and__num_Osimps_I6_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),Na)) ).

% and_num.simps(6)
tff(fact_9735_and__num_Osimps_I9_J,axiom,
    ! [M: num,Na: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tg,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),Na)) ).

% and_num.simps(9)
tff(fact_9736_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_9737_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_tl) ).

% Rats_eq_int_div_nat
tff(fact_9738_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),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)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9739_bij__betw__Suc,axiom,
    ! [M7: set_nat,N2: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M7),N2)
    <=> ( image_nat_nat(suc,M7) = N2 ) ) ).

% bij_betw_Suc
tff(fact_9740_Rats__abs__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(abs_abs_real,X),field_5140801741446780682s_real)
    <=> member_real(X,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_9741_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9742_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9743_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X2: real] :
      ( member_real(X2,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),X2) ) ).

% Rats_no_top_le
tff(fact_9744_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ? [X2: real] :
          ( member_real(X2,field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y) ) ) ).

% Rats_dense_in_real
tff(fact_9745_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X2: real] :
      ( member_real(X2,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),X) ) ).

% Rats_no_bot_less
tff(fact_9746_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9747_image__Suc__lessThan,axiom,
    ! [Na: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na) ).

% image_Suc_lessThan
tff(fact_9748_image__Suc__atMost,axiom,
    ! [Na: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Na)) ).

% image_Suc_atMost
tff(fact_9749_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9750_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9751_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9752_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_9753_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_tn) ).

% Rats_eq_int_div_int
tff(fact_9754_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9755_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : divide_divide_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% divide_rat
tff(fact_9756_floor__Fract,axiom,
    ! [A: int,B: int] : archim3151403230148437115or_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% floor_Fract
tff(fact_9757_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D))
        <=> aa_int_o(aa_int_fun_int_o(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),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),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),D))) ) ) ) ).

% less_rat
tff(fact_9758_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_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),D)) = 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),D)),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),D)) ) ) ) ).

% add_rat
tff(fact_9759_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(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),D))
        <=> aa_int_o(aa_int_fun_int_o(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),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),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),D))) ) ) ) ).

% le_rat
tff(fact_9760_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( minus_minus_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = 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),D)),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),D)) ) ) ) ).

% diff_rat
tff(fact_9761_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_9762_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_9763_Rat__induct__pos,axiom,
    ! [P: fun_rat_o,Q3: rat] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_rat_o(P,aa_int_rat(aa_int_fun_int_rat(fract,A4),B4)) )
     => aa_rat_o(P,Q3) ) ).

% Rat_induct_pos
tff(fact_9764_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_9765_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(aa_int_fun_int_rat(fract,A),B) = aa_int_rat(aa_int_fun_int_rat(fract,C),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9766_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_9767_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_9768_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_9769_Fract__coprime,axiom,
    ! [A: int,B: int] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B))) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ).

% Fract_coprime
tff(fact_9770_One__rat__def,axiom,
    one_one_rat = aa_int_rat(aa_int_fun_int_rat(fract,one_one_int),one_one_int) ).

% One_rat_def
tff(fact_9771_Fract__of__int__eq,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),one_one_int) = aa_int_rat(ring_1_of_int_rat,K) ).

% Fract_of_int_eq
tff(fact_9772_Fract__of__int__quotient,axiom,
    ! [K: int,L: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),L) = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K),aa_int_rat(ring_1_of_int_rat,L)) ).

% Fract_of_int_quotient
tff(fact_9773_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_9774_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) = numeral_numeral_rat(W) ).

% rat_number_collapse(3)
tff(fact_9775_rat__number__expand_I3_J,axiom,
    ! [K: num] : 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_9776_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9777_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9778_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9779_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% Fract_less_one_iff
tff(fact_9780_rat__number__collapse_I5_J,axiom,
    aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = uminus_uminus_rat(one_one_rat) ).

% rat_number_collapse(5)
tff(fact_9781_Fract__add__one,axiom,
    ! [Na: int,M: int] :
      ( ( Na != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),Na)),Na) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,M),Na)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9782_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_to(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_9783_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9784_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9785_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9786_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_9787_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) = uminus_uminus_rat(numeral_numeral_rat(W)) ).

% rat_number_collapse(4)
tff(fact_9788_rat__number__expand_I5_J,axiom,
    ! [K: num] : uminus_uminus_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_9789_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9790_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9791_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_ss)) ).

% less_int_def
tff(fact_9792_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_su)) ).

% less_eq_int_def
tff(fact_9793_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9794_positive__rat,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% positive_rat
tff(fact_9795_Rat_Opositive__add,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(positive,X)
     => ( aa_rat_o(positive,Y)
       => aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) ) ) ).

% Rat.positive_add
tff(fact_9796_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(positive,minus_minus_rat(Y,X)) ) ).

% less_rat_def
tff(fact_9797_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( aa_rat_o(positive,X)
    <=> aa_int_o(aa_int_fun_int_o(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_9798_Rat_Opositive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_tp) ).

% Rat.positive_def
tff(fact_9799_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_tq,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9800_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
     => ? [Q2: rat] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),field_7254667332652039916t_real(Q2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,field_7254667332652039916t_real(Q2)),Y) ) ) ).

% of_rat_dense
tff(fact_9801_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_tr) ).

% plus_rat_def
tff(fact_9802_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ts) ).

% inverse_rat_def
tff(fact_9803_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(one_one_int,one_one_int)) ).

% one_rat_def
tff(fact_9804_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      aa_int_rat(aa_int_fun_int_rat(fract,Xa),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,
        $ite(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_9805_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_9806_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,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,product_fst_int_int(Xa)),product_snd_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa)),product_snd_int_int(X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9807_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
            $ite(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_9808_ratrel__iff,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(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_9809_one__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,product_Pair_int_int(one_one_int,one_one_int)),product_Pair_int_int(one_one_int,one_one_int)) ).

% one_rat.rsp
tff(fact_9810_zero__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(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_9811_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_tu(Q3)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_9812_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_tw(Q3)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_9813_ratrel__def,axiom,
    ! [X3: product_prod_int_int,Xa2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X3),Xa2)
    <=> ( ( product_snd_int_int(X3) != 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(X3)),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(X3)) ) ) ) ).

% ratrel_def
tff(fact_9814_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_o(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X))
      <=> aa_int_o(aa_int_fun_int_o(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_9815_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_tx(D))) ).

% int_ge_less_than_def
tff(fact_9816_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ty(D))) ).

% int_ge_less_than2_def
tff(fact_9817_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)) ).

% min_Suc_Suc
tff(fact_9818_min__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Na) = zero_zero_nat ).

% min_0L
tff(fact_9819_min__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9820_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_9821_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% min_numeral_Suc
tff(fact_9822_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9823_nat__mult__min__right,axiom,
    ! [M: nat,Na: 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,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_min_right
tff(fact_9824_nat__mult__min__left,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),Q3)) ).

% nat_mult_min_left
tff(fact_9825_min__diff,axiom,
    ! [M: nat,I: nat,Na: 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,Na),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),I) ).

% min_diff
tff(fact_9826_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9827_concat__bit__assoc__sym,axiom,
    ! [M: nat,Na: nat,K: int,L: int,R4: int] : aa_int_int(bit_concat_bit(M,aa_int_int(bit_concat_bit(Na,K),L)),R4) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),L),R4)) ).

% concat_bit_assoc_sym
tff(fact_9828_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Na: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_concat_bit(Na,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9829_min__Suc1,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_tz,Na)),M) ).

% min_Suc1
tff(fact_9830_min__Suc2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ua,Na)),M) ).

% min_Suc2
tff(fact_9831_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(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_9832_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ub(M),Na)))) ).

% divide_nat_def
tff(fact_9833_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_uc(Na),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9834_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_9835_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_9836_Max__divisors__self__int,axiom,
    ! [Na: int] :
      ( ( Na != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ag,Na))) = aa_int_int(abs_abs_int,Na) ) ) ).

% Max_divisors_self_int
tff(fact_9837_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9838_gcd__is__Max__divisors__int,axiom,
    ! [Na: int,M: int] :
      ( ( Na != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),Na) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ud(Na),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9839_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M7)
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(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_ue,M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9840_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_9841_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9842_Inf__real__def,axiom,
    ! [X6: set_real] : comple4887499456419720421f_real(X6) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X6))) ).

% Inf_real_def
tff(fact_9843_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_uf(X6)) ).

% Sup_int_def
tff(fact_9844_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X6) = $ite(X6 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X6)) ).

% Sup_nat_def
tff(fact_9845_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ug,Na),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na) ) ) ).

% range_mod
tff(fact_9846_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_9847_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9848_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9849_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9850_infinite__UNIV__int,axiom,
    ~ finite_finite_int(top_top_set_int) ).

% infinite_UNIV_int
tff(fact_9851_int__in__range__abs,axiom,
    ! [Na: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,Na),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9852_root__def,axiom,
    ! [Na: nat,X: real] :
      aa_real_real(aa_nat_fun_real_real(root,Na),X) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_uh,Na)),X)) ).

% root_def
tff(fact_9853_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_9854_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_9855_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_size_char(char2((X1),(X22),(X32),(X42),(X52),(X62),(X72),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9856_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(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_9857_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_9858_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B7: $o] : 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,aa_o_Code_integer(zero_n356916108424825756nteger,(B7))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B62)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B52)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B42)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B32)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B22)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B1)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_o_Code_integer(zero_n356916108424825756nteger,(B0))) ).

% integer_of_char_code
tff(fact_9859_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_char(char2((X1),(X22),(X32),(X42),(X52),(X62),(X72),(X8))) = zero_zero_nat ).

% char.size_gen
tff(fact_9860_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_9861_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9862_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9863_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_9864_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_9865_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X2: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X2,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9866_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9867_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : nat_list_encode(cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_9868_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),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_9869_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))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa)) ) ) ) ).

% upto.pelims
tff(fact_9870_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9871_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9872_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9873_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9874_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9875_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9876_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_9877_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),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,Na))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9878_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),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,Na)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9879_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)),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,Na))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9880_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      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,Na))) = $ite(aa_int_o(aa_int_fun_int_o(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,Na))),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,Na)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9881_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9882_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_9883_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_9884_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9885_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(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_9886_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(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_9887_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9888_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9889_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) ) ) ).

% upto.elims
tff(fact_9890_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9891_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9892_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9893_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9894_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(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_9895_DERIV__even__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),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,Na))),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9896_DERIV__real__root__generic,axiom,
    ! [Na: nat,X: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( X != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9897_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
            & aa_real_o(aa_real_fun_real_o(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(F2,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9898_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2))),D)
             => ( aa_real_real(F,X) = aa_real_real(F,Y2) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9899_DERIV__ln,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_9900_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(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_9901_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(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_9902_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9903_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(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_9904_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(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_9905_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( member_real(X,set_or1633881224788618240n_real(A,B))
       => ( member_real(Y,set_or1633881224788618240n_real(A,B))
         => ( ! [X2: real] :
                ( member_real(X2,set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9906_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(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_9907_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(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_9908_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(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_9909_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(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_9910_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9911_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9912_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(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))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9913_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9914_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,X)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9915_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(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_9916_DERIV__pow,axiom,
    ! [Na: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ui,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_9917_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X: real,Na: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_uj(G),Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9918_has__real__derivative__powr,axiom,
    ! [Z: real,R4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_uk,R4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R4),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9919_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(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_9920_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R4: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ul(G),R4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R4),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9921_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_real,R4: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => ( has_fi5821293074295781190e_real(F,R4,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_um(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,R4),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_9922_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9923_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(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_9924_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_9925_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
           => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9926_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9927_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(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(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9928_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_un(F),X2)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_up(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_un(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9929_DERIV__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9930_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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(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_9931_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(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(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_9932_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M4: nat,X2: 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)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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_uq(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9933_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,X2: 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)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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_uq(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9934_DERIV__odd__real__root,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9935_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_ma(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9936_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ? [T: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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_ma(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9937_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M4: nat,T: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_ma(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9938_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X2: 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)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T)),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_uq(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9939_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M4: nat,T: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),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_uq(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(X),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9940_Taylor__down,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_ur(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9941_Taylor__up,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_ur(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9942_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M4: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
                 => ( ( X != C )
                   => ? [T: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,X),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_us(Diff),C),X)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),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,Na),T)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9943_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M4: nat,T: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),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)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M2: nat,T2: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_uu(Na),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2)),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_uv(Diff),M2),T2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(T2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T2,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9944_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_uw,suminf_real(aTP_Lamm_ux(X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9945_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R2) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9946_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 )
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R2) )
               => ( aa_real_real(F,X3) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9947_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))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R2) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9948_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_uy,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9949_binomial__def,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_uz(Na),K))) ).

% binomial_def
tff(fact_9950_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_va(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),N7))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9951_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_va(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),N7)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9952_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vb,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9953_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_9954_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_vc,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9955_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vd(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L2)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9956_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R2: real] :
        ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_nat_real(X6,N)) )
     => filterlim_nat_real(aTP_Lamm_ve(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9957_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_vf,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9958_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aTP_Lamm_vg(C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9959_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_vh,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9960_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R4: real] : filterlim_nat_real(aTP_Lamm_vi(R4),topolo2815343760600316023s_real(R4),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9961_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N7)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9962_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => filterlim_nat_real(power_power_real(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9963_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_vj(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9964_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(power_power_real(C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9965_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_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_9966_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aTP_Lamm_vk(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9967_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R4: real] : filterlim_nat_real(aTP_Lamm_vl(R4),topolo2815343760600316023s_real(R4),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9968_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aTP_Lamm_vm(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9969_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R4: real] : filterlim_nat_real(aTP_Lamm_vn(R4),topolo2815343760600316023s_real(R4),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9970_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_va(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9971_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_va(A)) ) ) ) ).

% summable
tff(fact_9972_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_vo(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K3: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_vp(Theta),K3),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9973_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_vq(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K3: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_vp(Theta),K3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9974_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_vr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_va(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9975_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => filterlim_nat_real(aTP_Lamm_bj(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9976_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),Na)))),suminf_real(aTP_Lamm_va(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9977_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_vr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_va(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9978_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),N7)))),L2)
              & filterlim_nat_real(aTP_Lamm_vr(A),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_vs(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9979_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_vs(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_va(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9980_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_vs(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_va(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9981_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_va(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(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))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9982_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_vt(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9983_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_vu(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9984_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X2)
         => aa_nat_o(P,X2) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9985_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N8: nat] :
        ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
         => aa_nat_o(P,N4) ) ) ).

% eventually_sequentially
tff(fact_9986_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_9987_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_vu(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9988_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9989_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9990_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F2),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_9991_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F2),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_9992_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_vx(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9993_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9994_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_vy(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9995_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9996_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_9997_artanh__real__at__left__1,axiom,
    filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).

% artanh_real_at_left_1
tff(fact_9998_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_wa,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9999_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_wb,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10000_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F2: 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_vy(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10001_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F2: 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_vy(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),F3,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),F3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10002_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_wc,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10003_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10004_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X2)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10005_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10006_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_wd(Na),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10007_tanh__real__at__bot,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).

% tanh_real_at_bot
tff(fact_10008_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F2),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_10009_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_10010_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_wd(Na),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10011_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F2),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_10012_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10013_Bseq__realpow,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => bfun_nat_real(power_power_real(X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10014_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_we,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_10015_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10016_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_wf(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10017_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_vx(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_10018_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_10019_artanh__real__at__right__1,axiom,
    filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).

% artanh_real_at_right_1
tff(fact_10020_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F2),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_10021_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F2: 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_vy(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10022_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F2: 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_vy(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F0),F2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10023_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F2),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_10024_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_vy(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10025_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F2: 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_vy(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(F),F2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vv(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G2),F2),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10026_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_10027_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_10028_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_10029_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_10030_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_10031_isCont__real__sqrt,axiom,
    ! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_10032_isCont__real__root,axiom,
    ! [X: real,Na: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),aa_nat_fun_real_real(root,Na)) ).

% isCont_real_root
tff(fact_10033_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(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_10034_isCont__arcosh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_10035_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
           => ( ! [Y2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),B)
                   => ( aa_real_real(F,aa_real_real(G,Y2)) = Y2 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_10036_isCont__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_10037_isCont__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_10038_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),X)
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(B,X))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ).

% LIM_less_bound
tff(fact_10039_isCont__artanh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_10040_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X))),D)
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),X))),D)
             => 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_10041_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),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,C3)) = 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(F2,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10042_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G) )
           => ( ! [X2: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X2,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),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_10043_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ? [L2: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
              & has_fi5821293074295781190e_real(F,L2,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)),L2) ) ) ) ) ) ).

% MVT
tff(fact_10044_mono__times__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na)) ) ).

% mono_times_nat
tff(fact_10045_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10046_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A2,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10047_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X2)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_wg(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10048_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_10049_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_10050_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A2,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10051_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => member_real(aa_real_real(F,X2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_wh(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10052_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F2: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & ! [X3: real] : aa_real_real(aa_rea7340605660300375871l_real(F2,Z2),X3) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_10053_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F2: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
               => ( aa_real_o(aa_real_fun_real_o(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(F2,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_10054_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_wi,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_10055_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_10056_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_10057_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_10058_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10059_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_10060_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_uh,Na),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10061_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => inj_on_real_real(aa_rea7340605660300375871l_real(log,B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_10062_inj__on__diff__nat,axiom,
    ! [N2: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,K)),N2) ) ).

% inj_on_diff_nat
tff(fact_10063_inj__Suc,axiom,
    ! [N2: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N2) ).

% inj_Suc
tff(fact_10064_inj__on__char__of__nat,axiom,
    aa_set_nat_o(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_10065_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_10066_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_10067_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_10068_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_10069_powr__real__of__int_H,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( ( X != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,Na)) = aa_int_real(power_int_real(X),Na) ) ) ) ).

% powr_real_of_int'
tff(fact_10070_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wj)) ).

% pred_nat_def
tff(fact_10071_less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),Na),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% less_eq
tff(fact_10072_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),Na),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% pred_nat_trancl_eq_le
tff(fact_10073_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F2: fun_real_real] :
      ( ! [X2: real] : has_fi5821293074295781190e_real(F,aa_real_real(F2,X2),topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( ! [X2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F2,X2))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_10074_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(F,Na)) ) ).

% strict_mono_imp_increasing
tff(fact_10075_rat__floor__code,axiom,
    ! [P3: rat] : archim3151403230148437115or_rat(P3) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P3)) ).

% rat_floor_code
tff(fact_10076_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_wk,Qr) ).

% Divides.adjust_div_def
tff(fact_10077_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),X)
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X2: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X2,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X2,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_10078_remdups__upt,axiom,
    ! [M: nat,Na: nat] : remdups_nat(upt(M,Na)) = upt(M,Na) ).

% remdups_upt
tff(fact_10079_tl__upt,axiom,
    ! [M: nat,Na: nat] : tl_nat(upt(M,Na)) = upt(aa_nat_nat(suc,M),Na) ).

% tl_upt
tff(fact_10080_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_10081_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).

% drop_upt
tff(fact_10082_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_10083_take__upt,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),Na)
     => ( take_nat(M,upt(I,Na)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_10084_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_10085_sorted__list__of__set__range,axiom,
    ! [M: nat,Na: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = upt(M,Na) ).

% sorted_list_of_set_range
tff(fact_10086_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_10087_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_10088_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)),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,Na))),nil_nat) ).

% upt_rec_numeral
tff(fact_10089_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(suc,upt(M,Na)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na)) ).

% map_Suc_upt
tff(fact_10090_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).

% distinct_upt
tff(fact_10091_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M) = set_nat2(upt(Na,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_10092_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Na),M) = set_nat2(upt(aa_nat_nat(suc,Na),M)) ).

% greaterThanLessThan_upt
tff(fact_10093_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_10094_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Na),M) = set_nat2(upt(aa_nat_nat(suc,Na),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_10095_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(M,cons_nat(Na,Ns)) = upt(M,Q3) )
    <=> ( cons_nat(Na,Ns) = upt(aa_nat_nat(suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_10096_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_10097_atLeast__upt,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Na) = set_nat2(upt(zero_zero_nat,Na)) ).

% atLeast_upt
tff(fact_10098_atMost__upto,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,Na) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Na))) ).

% atMost_upto
tff(fact_10099_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_10100_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_10101_map__add__upt,axiom,
    ! [Na: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,Na),upt(zero_zero_nat,M)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% map_add_upt
tff(fact_10102_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(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_10103_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_10104_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(aTP_Lamm_wm,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_10105_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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_10106_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_10107_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups4561878855575611511st_nat(upt(M,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) ) ) ).

% sum_list_upt
tff(fact_10108_card__length__sum__list__rec,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_wn(M),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_wo(M),N2)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_wp(M),N2)))) ) ) ).

% card_length_sum_list_rec
tff(fact_10109_card__length__sum__list,axiom,
    ! [M: nat,N2: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_wn(M),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),one_one_nat)),N2) ).

% card_length_sum_list
tff(fact_10110_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_10111_sorted__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(M,Na)) ).

% sorted_upt
tff(fact_10112_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_10113_sorted__upto,axiom,
    ! [M: int,Na: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(M,Na)) ).

% sorted_upto
tff(fact_10114_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_10115_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_wq(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_wr(M)) ).

% pairs_le_eq_Sigma
tff(fact_10116_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_ws(X)) ).

% quotient_of_def
tff(fact_10117_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_int_int(abs_abs_int,K)),L)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),L) ) ).

% coprime_abs_left_iff
tff(fact_10118_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),aa_int_int(abs_abs_int,L))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),L) ) ).

% coprime_abs_right_iff
tff(fact_10119_normalize__stable,axiom,
    ! [Q3: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P3),Q3)
       => ( normalize(product_Pair_int_int(P3,Q3)) = product_Pair_int_int(P3,Q3) ) ) ) ).

% normalize_stable
tff(fact_10120_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),B)
         => ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_10121_Rat__cases,axiom,
    ! [Q3: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q3 = aa_int_rat(aa_int_fun_int_rat(fract,A4),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) ).

% Rat_cases
tff(fact_10122_Rat__induct,axiom,
    ! [P: fun_rat_o,Q3: rat] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4)
           => aa_rat_o(P,aa_int_rat(aa_int_fun_int_rat(fract,A4),B4)) ) )
     => aa_rat_o(P,Q3) ) ).

% Rat_induct
tff(fact_10123_Rat__cases__nonzero,axiom,
    ! [Q3: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q3 = aa_int_rat(aa_int_fun_int_rat(fract,A4),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ( ( A4 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) )
     => ( Q3 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_10124_quotient__of__unique,axiom,
    ! [R4: rat] :
    ? [X2: product_prod_int_int] :
      ( ( R4 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(X2)),product_snd_int_int(X2)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(X2))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(X2)),product_snd_int_int(X2))
      & ! [Y3: product_prod_int_int] :
          ( ( ( R4 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Y3)),product_snd_int_int(Y3)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Y3))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Y3)),product_snd_int_int(Y3)) )
         => ( Y3 = X2 ) ) ) ).

% quotient_of_unique
tff(fact_10125_coprime__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M),Na) ) ).

% coprime_int_iff
tff(fact_10126_coprime__nat__abs__right__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_10127_coprime__nat__abs__left__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% coprime_nat_abs_left_iff
tff(fact_10128_coprime__Suc__0__left,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ).

% coprime_Suc_0_left
tff(fact_10129_coprime__Suc__0__right,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_10130_coprime__Suc__right__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(suc,Na)) ).

% coprime_Suc_right_nat
tff(fact_10131_coprime__Suc__left__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Na)),Na) ).

% coprime_Suc_left_nat
tff(fact_10132_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),B)
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_10133_coprime__diff__one__right__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_10134_coprime__diff__one__left__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),Na) ) ).

% coprime_diff_one_left_nat
tff(fact_10135_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( member_real(X,field_5140801741446780682s_real)
     => ~ ! [M4: nat,N: nat] :
            ( ( N != 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,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M4),N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_10136_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_10137_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_10138_sort__upt,axiom,
    ! [M: nat,Na: nat] : linord738340561235409698at_nat(aTP_Lamm_am,upt(M,Na)) = upt(M,Na) ).

% sort_upt
tff(fact_10139_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_an,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_10140_Rat_Opositive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_tp,aTP_Lamm_tp) ).

% Rat.positive.rsp
tff(fact_10141_vanishes__mult__bounded,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( ? [A6: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A6)
          & ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_nat_rat(X6,N))),A6) )
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wt,X6),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_10142_vanishes__const,axiom,
    ! [C: rat] :
      ( vanishes(aTP_Lamm_wu(C))
    <=> ( C = zero_zero_rat ) ) ).

% vanishes_const
tff(fact_10143_vanishes__minus,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
     => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_wv,X6)) ) ).

% vanishes_minus
tff(fact_10144_vanishes__add,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( vanishes(X6)
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ww,X6),Y5)) ) ) ).

% vanishes_add
tff(fact_10145_vanishes__diff,axiom,
    ! [X6: fun_nat_rat,Y5: fun_nat_rat] :
      ( vanishes(X6)
     => ( vanishes(Y5)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_wx(X6),Y5)) ) ) ).

% vanishes_diff
tff(fact_10146_vanishesD,axiom,
    ! [X6: fun_nat_rat,R4: rat] :
      ( vanishes(X6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R4)
       => ? [K3: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N7)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_nat_rat(X6,N7))),R4) ) ) ) ).

% vanishesD
tff(fact_10147_vanishesI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
         => ? [K4: nat] :
            ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_nat_rat(X6,N))),R2) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_10148_vanishes__def,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K2: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_nat_rat(X6,N4))),R5) ) ) ) ).

% vanishes_def
tff(fact_10149_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 )
             => ! [N: num] :
                  ( ( Xa = aa_num_num(bit0,N) )
                 => ( ( Y = some_num(one) )
                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa = aa_num_num(bit1,N) )
                   => ( ( Y = none_num )
                     => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
             => ( ! [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) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit0,N) )
                         => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M4,N)) )
                           => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N: num] :
                            ( ( Xa = aa_num_num(bit1,N) )
                           => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M4,N)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N))) ) ) )
                   => ( ! [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) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit0,N) )
                               => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tg,bit_and_not_num(M4,N)) )
                                 => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N: num] :
                                  ( ( Xa = aa_num_num(bit1,N) )
                                 => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M4,N)) )
                                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_10150_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_tr,aTP_Lamm_tr) ).

% plus_rat.rsp
tff(fact_10151_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_wy,aTP_Lamm_wy) ).

% Fract.rsp
tff(fact_10152_less__eq__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_10153_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_10154_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_10155_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_10156_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_wz,aTP_Lamm_wz) ).

% sub.rsp
tff(fact_10157_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_xa),aTP_Lamm_xa) ).

% dup.rsp
tff(fact_10158_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_10159_plus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).

% plus_integer.rsp
tff(fact_10160_Suc_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc) ).

% Suc.rsp
tff(fact_10161_divide__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),divide_divide_nat,divide_divide_nat) ).

% divide_natural.rsp
tff(fact_10162_divide__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),divide_divide_int,divide_divide_int) ).

% divide_integer.rsp
tff(fact_10163_inverse__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ts),aTP_Lamm_ts) ).

% inverse_rat.rsp
tff(fact_10164_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 )
             => ! [N: num] :
                  ( ( Xa = aa_num_num(bit0,N) )
                 => ( ( Y = none_num )
                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa = aa_num_num(bit1,N) )
                   => ( ( Y = some_num(one) )
                     => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
             => ( ! [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) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit0,N) )
                         => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N: num] :
                            ( ( Xa = aa_num_num(bit1,N) )
                           => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N))) ) ) )
                   => ( ! [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) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit0,N) )
                               => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N: num] :
                                  ( ( Xa = aa_num_num(bit1,N) )
                                 => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tg,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M4),N)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_10165_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 )
             => ! [N: num] :
                  ( ( Xa = aa_num_num(bit0,N) )
                 => ( ( Y = some_num(aa_num_num(bit1,N)) )
                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N))) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa = aa_num_num(bit1,N) )
                   => ( ( Y = some_num(aa_num_num(bit0,N)) )
                     => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N))) ) ) )
             => ( ! [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) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit0,N) )
                         => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N)) )
                           => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa_num_num(bit0,M4) )
                       => ! [N: num] :
                            ( ( Xa = aa_num_num(bit1,N) )
                           => ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N))) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N))) ) ) )
                   => ( ! [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) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit0,N) )
                               => ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N))) )
                                 => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa_num_num(bit1,M4) )
                             => ! [N: num] :
                                  ( ( Xa = aa_num_num(bit1,N) )
                                 => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M4),N)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_10166_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: 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))) ) ) )
             => ( ! [N: num] :
                    ( ( X = aa_num_num(bit0,N) )
                   => ( ( Xa = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N),one)) ) ) )
               => ( ! [N: num] :
                      ( ( X = aa_num_num(bit0,N) )
                     => ! [M4: num] :
                          ( ( Xa = aa_num_num(bit0,M4) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N,M4)) )
                           => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit0,M4))) ) ) )
                 => ( ! [N: num] :
                        ( ( X = aa_num_num(bit0,N) )
                       => ! [M4: num] :
                            ( ( Xa = aa_num_num(bit1,M4) )
                           => ( ( Y = aa_num_num(bit0,bit_or_not_num_neg(N,M4)) )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit1,M4))) ) ) )
                   => ( ! [N: num] :
                          ( ( X = aa_num_num(bit1,N) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N),one)) ) ) )
                     => ( ! [N: num] :
                            ( ( X = aa_num_num(bit1,N) )
                           => ! [M4: num] :
                                ( ( Xa = aa_num_num(bit0,M4) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N,M4)) )
                                 => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit0,M4))) ) ) )
                       => ~ ! [N: num] :
                              ( ( X = aa_num_num(bit1,N) )
                             => ! [M4: num] :
                                  ( ( Xa = aa_num_num(bit1,M4) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N,M4)) )
                                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit1,M4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_10167_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_10168_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_10169_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_tr,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_10170_one__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,product_Pair_int_int(one_one_int,one_one_int)),one_one_rat) ).

% one_rat.transfer
tff(fact_10171_zero__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,product_Pair_int_int(zero_zero_int,one_one_int)),zero_zero_rat) ).

% zero_rat.transfer
tff(fact_10172_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_wy,fract) ).

% Fract.transfer
tff(fact_10173_Rat_Opositive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_tp,positive) ).

% Rat.positive.transfer
tff(fact_10174_inverse__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ts),inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_10175_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_sp),times_times_int) ).

% times_int.transfer
tff(fact_10176_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_10177_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_xb,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_10178_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_sq)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_10179_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_10180_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_10181_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_ss),ord_less_int) ).

% less_int.transfer
tff(fact_10182_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_su),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_10183_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_sw),plus_plus_int) ).

% plus_int.transfer
tff(fact_10184_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_sy),minus_minus_int) ).

% minus_int.transfer
tff(fact_10185_inverse__real_Otransfer,axiom,
    aa_fun_real_real_o(aa_fun1584707981518017559real_o(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_xe),inverse_inverse_real) ).

% inverse_real.transfer
tff(fact_10186_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_sp),produc4528391901023682385at_nat(aTP_Lamm_sp)) ).

% times_int.rsp
tff(fact_10187_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(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_10188_int_Orel__eq__transfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_10189_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_10190_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_sq)),produc2626176000494625587at_nat(aTP_Lamm_sq)) ).

% uminus_int.rsp
tff(fact_10191_zero__real_Otransfer,axiom,
    aa_real_o(aa_fun8455130352500885026real_o(pcr_real,aTP_Lamm_xc),zero_zero_real) ).

% zero_real.transfer
tff(fact_10192_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(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_10193_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) )
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,X),Y) ) ).

% int.abs_eq_iff
tff(fact_10194_one__real_Otransfer,axiom,
    aa_real_o(aa_fun8455130352500885026real_o(pcr_real,aTP_Lamm_xf),one_one_real) ).

% one_real.transfer
tff(fact_10195_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(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_10196_uminus__real_Otransfer,axiom,
    aa_fun_real_real_o(aa_fun1584707981518017559real_o(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_wv),uminus_uminus_real) ).

% uminus_real.transfer
tff(fact_10197_plus__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_ww,plus_plus_real) ).

% plus_real.transfer
tff(fact_10198_times__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_wt,times_times_real) ).

% times_real.transfer
tff(fact_10199_intrel__def,axiom,
    intrel = produc6395135316500328296_nat_o(aTP_Lamm_xh) ).

% intrel_def
tff(fact_10200_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_ss),produc6395135316500328296_nat_o(aTP_Lamm_ss)) ).

% less_int.rsp
tff(fact_10201_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_su),produc6395135316500328296_nat_o(aTP_Lamm_su)) ).

% less_eq_int.rsp
tff(fact_10202_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_sw),produc4528391901023682385at_nat(aTP_Lamm_sw)) ).

% plus_int.rsp
tff(fact_10203_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_sy),produc4528391901023682385at_nat(aTP_Lamm_sy)) ).

% minus_int.rsp
tff(fact_10204_Real_Opositive_Otransfer,axiom,
    bNF_re3218747500659316912al_o_o(pcr_real,fequal_o,aTP_Lamm_xi,positive2) ).

% Real.positive.transfer
tff(fact_10205_rcis__inverse,axiom,
    ! [R4: real,A: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(rcis(R4),A)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),R4)),aa_real_real(uminus_uminus_real,A)) ).

% rcis_inverse
tff(fact_10206_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
    <=> aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X)) ) ).

% less_real_def
tff(fact_10207_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ aa_real_o(positive2,X)
     => ( ( X != zero_zero_real )
       => aa_real_o(positive2,aa_real_real(uminus_uminus_real,X)) ) ) ).

% Real.positive_minus
tff(fact_10208_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_ts,Uu) = $ite(product_fst_int_int(Uu) = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(product_snd_int_int(Uu),product_fst_int_int(Uu))) ).

% ATP.lambda_1
tff(fact_10209_ATP_Olambda__2,axiom,
    ! [Uu: fun_nat_rat] :
      aa_fun765758790203053448at_rat(aTP_Lamm_xe,Uu) = $ite(vanishes(Uu),aTP_Lamm_xc,aTP_Lamm_xd(Uu)) ).

% ATP.lambda_2
tff(fact_10210_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_bu,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uu)),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,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_3
tff(fact_10211_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_dh,Uu) = aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_4
tff(fact_10212_ATP_Olambda__5,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ri,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),numeral_numeral_real(aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_5
tff(fact_10213_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ku,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(power_power_real(zero_zero_real),Uu)) ).

% ATP.lambda_6
tff(fact_10214_ATP_Olambda__7,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_uy,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_7
tff(fact_10215_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_wa,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_8
tff(fact_10216_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_vc,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_9
tff(fact_10217_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_wm,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_10
tff(fact_10218_ATP_Olambda__11,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_xa,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_11
tff(fact_10219_ATP_Olambda__12,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_eu,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_12
tff(fact_10220_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_er,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_13
tff(fact_10221_ATP_Olambda__14,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ev,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_14
tff(fact_10222_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_et,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_15
tff(fact_10223_ATP_Olambda__16,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_es,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_16
tff(fact_10224_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_xb,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_17
tff(fact_10225_ATP_Olambda__18,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_tp,Uu)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu))) ) ).

% ATP.lambda_18
tff(fact_10226_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_vf,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_19
tff(fact_10227_ATP_Olambda__20,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_uw,Uu) = suminf_real(aTP_Lamm_bl(Uu)) ).

% ATP.lambda_20
tff(fact_10228_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_ue,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,Uu)) ).

% ATP.lambda_21
tff(fact_10229_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_vh,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_22
tff(fact_10230_ATP_Olambda__23,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_tg,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_23
tff(fact_10231_ATP_Olambda__24,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_tc,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_24
tff(fact_10232_ATP_Olambda__25,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_rl,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_25
tff(fact_10233_ATP_Olambda__26,axiom,
    ! [Uu: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aTP_Lamm_xi,Uu)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K2: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Uu,N4)) ) ) ) ).

% ATP.lambda_26
tff(fact_10234_ATP_Olambda__27,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_tl,Uu)
    <=> ? [I3: int,N4: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_27
tff(fact_10235_ATP_Olambda__28,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_tn,Uu)
    <=> ? [I3: int,J3: int] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_28
tff(fact_10236_ATP_Olambda__29,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_te,Uu) = some_num(one) ).

% ATP.lambda_29
tff(fact_10237_ATP_Olambda__30,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tj(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_th(Uua),aTP_Lamm_ti(Uua),Uu) ).

% ATP.lambda_30
tff(fact_10238_ATP_Olambda__31,axiom,
    ! [Uu: complex,Uua: nat] :
      aa_nat_complex(aTP_Lamm_eo(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_31
tff(fact_10239_ATP_Olambda__32,axiom,
    ! [Uu: real,Uua: nat] :
      aa_nat_real(aTP_Lamm_en(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua)),zero_zero_real) ).

% ATP.lambda_32
tff(fact_10240_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_ji(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_33
tff(fact_10241_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jk,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_34
tff(fact_10242_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_jj(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_35
tff(fact_10243_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jh,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_36
tff(fact_10244_ATP_Olambda__37,axiom,
    ! [Uu: complex,Uua: nat] :
      aa_nat_complex(aTP_Lamm_eq(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua))) ).

% ATP.lambda_37
tff(fact_10245_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: nat] :
      aa_nat_real(aTP_Lamm_ep(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua))) ).

% ATP.lambda_38
tff(fact_10246_ATP_Olambda__39,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_di(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,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_39
tff(fact_10247_ATP_Olambda__40,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rw,Uu),Uua) = $ite(Uu = 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,Uu)),Uua),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_40
tff(fact_10248_ATP_Olambda__41,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wy,Uu),Uua) = $ite(Uua = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Uu,Uua)) ).

% ATP.lambda_41
tff(fact_10249_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_je(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_42
tff(fact_10250_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jg,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_43
tff(fact_10251_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_jf(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_44
tff(fact_10252_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jd,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(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,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_45
tff(fact_10253_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_tk,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_tj(Uua),Uu) ).

% ATP.lambda_46
tff(fact_10254_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_th(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_tc,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_47
tff(fact_10255_ATP_Olambda__48,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_td(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_tc,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_48
tff(fact_10256_ATP_Olambda__49,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_tr,Uu),Uua) = 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,product_fst_int_int(Uu)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_49
tff(fact_10257_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_50
tff(fact_10258_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_cw(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_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(power_power_real(Uu),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_51
tff(fact_10259_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_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(power_power_real(Uu),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_52
tff(fact_10260_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_im(Uu),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,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_10261_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_io(Uu),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,Uu),Uua))) ).

% ATP.lambda_54
tff(fact_10262_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_iq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_10263_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ip(Uu),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_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,Uu),Uua))) ).

% ATP.lambda_56
tff(fact_10264_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_in,Uu),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,Uu),Uua))) ).

% ATP.lambda_57
tff(fact_10265_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ir(Uu),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_58
tff(fact_10266_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_it,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(power_power_real(numeral_numeral_real(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_59
tff(fact_10267_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_is(Uu),Uua) = divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_60
tff(fact_10268_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_rk,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_61
tff(fact_10269_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_os,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_62
tff(fact_10270_ATP_Olambda__63,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_sn,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_63
tff(fact_10271_ATP_Olambda__64,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_ew(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_64
tff(fact_10272_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_rs(Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),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_65
tff(fact_10273_ATP_Olambda__66,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_rv(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_66
tff(fact_10274_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bl(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(Uu),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_67
tff(fact_10275_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ux(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(power_power_real(Uu),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_68
tff(fact_10276_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_va(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_69
tff(fact_10277_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_iy(Uu),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,Uu),Uua))) ).

% ATP.lambda_70
tff(fact_10278_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ja(Uu),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,Uu),Uua))) ).

% ATP.lambda_71
tff(fact_10279_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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,Uu),Uua))) ).

% ATP.lambda_72
tff(fact_10280_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_jb(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_73
tff(fact_10281_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_iz,Uu),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,Uu),Uua))) ).

% ATP.lambda_74
tff(fact_10282_ATP_Olambda__75,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ws(Uu),Uua)
    <=> ( ( Uu = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Uua))
        & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua)) ) ) ).

% ATP.lambda_75
tff(fact_10283_ATP_Olambda__76,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gc(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_76
tff(fact_10284_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ge(Uu),Uua) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_77
tff(fact_10285_ATP_Olambda__78,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gd(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_78
tff(fact_10286_ATP_Olambda__79,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_jl(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_79
tff(fact_10287_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_jn(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_80
tff(fact_10288_ATP_Olambda__81,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jm(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),minus_minus_rat(divide_divide_rat(Uu,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_81
tff(fact_10289_ATP_Olambda__82,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gw(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)) ).

% ATP.lambda_82
tff(fact_10290_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_gy(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(Uu),Uua)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_83
tff(fact_10291_ATP_Olambda__84,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gx(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Uua)) ).

% ATP.lambda_84
tff(fact_10292_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sl,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_85
tff(fact_10293_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_86
tff(fact_10294_ATP_Olambda__87,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_uf(Uu),Uua)
    <=> ( member_int(Uua,Uu)
        & ! [X4: int] :
            ( member_int(X4,Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Uua) ) ) ) ).

% ATP.lambda_87
tff(fact_10295_ATP_Olambda__88,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_mi(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_88
tff(fact_10296_ATP_Olambda__89,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_wb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_89
tff(fact_10297_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_90
tff(fact_10298_ATP_Olambda__91,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_91
tff(fact_10299_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ga,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_92
tff(fact_10300_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ko(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_93
tff(fact_10301_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_vm(Uu),Uua) = aa_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,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_94
tff(fact_10302_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_we,Uu),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,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_95
tff(fact_10303_ATP_Olambda__96,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_wc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_96
tff(fact_10304_ATP_Olambda__97,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bj(Uu),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(power_power_real(Uu),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_97
tff(fact_10305_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qs(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,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(Uu,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_98
tff(fact_10306_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qt(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,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(Uu,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_99
tff(fact_10307_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qu(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,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(Uu,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_100
tff(fact_10308_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qv(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,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(Uu,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_101
tff(fact_10309_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gl(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,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(Uu,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_102
tff(fact_10310_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gi(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,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(Uu,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_103
tff(fact_10311_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gk(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,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(Uu,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_104
tff(fact_10312_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gj(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,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(Uu,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_105
tff(fact_10313_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_le(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_106
tff(fact_10314_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lc(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_107
tff(fact_10315_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ld(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_108
tff(fact_10316_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bm(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_109
tff(fact_10317_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_bn(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_110
tff(fact_10318_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cn(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_111
tff(fact_10319_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nu(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_112
tff(fact_10320_ATP_Olambda__113,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ns(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_113
tff(fact_10321_ATP_Olambda__114,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nt(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_114
tff(fact_10322_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_no(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_115
tff(fact_10323_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_116
tff(fact_10324_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_np(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_117
tff(fact_10325_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nq(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_118
tff(fact_10326_ATP_Olambda__119,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ft(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_119
tff(fact_10327_ATP_Olambda__120,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_fr(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_120
tff(fact_10328_ATP_Olambda__121,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_fs(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_121
tff(fact_10329_ATP_Olambda__122,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_po(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_122
tff(fact_10330_ATP_Olambda__123,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pq(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_123
tff(fact_10331_ATP_Olambda__124,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ng(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_124
tff(fact_10332_ATP_Olambda__125,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nd(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_125
tff(fact_10333_ATP_Olambda__126,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vs(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(Uu)),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,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vr(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(Uu)),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,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_128
tff(fact_10336_ATP_Olambda__129,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ds(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_129
tff(fact_10337_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_130
tff(fact_10338_ATP_Olambda__131,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_em(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_131
tff(fact_10339_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_el(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_132
tff(fact_10340_ATP_Olambda__133,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_du(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_133
tff(fact_10341_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_134
tff(fact_10342_ATP_Olambda__135,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bs(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_135
tff(fact_10343_ATP_Olambda__136,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_136
tff(fact_10344_ATP_Olambda__137,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_dm(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_137
tff(fact_10345_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dl(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_138
tff(fact_10346_ATP_Olambda__139,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_tb(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),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,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_139
tff(fact_10347_ATP_Olambda__140,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ei(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(cos_coeff,Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uu)),Uua)) ).

% ATP.lambda_140
tff(fact_10348_ATP_Olambda__141,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_eh(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uu)),Uua)) ).

% ATP.lambda_141
tff(fact_10349_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_uh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_142
tff(fact_10350_ATP_Olambda__143,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_dw(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(sin_coeff,Uua)),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_143
tff(fact_10351_ATP_Olambda__144,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dv(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_144
tff(fact_10352_ATP_Olambda__145,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_dy(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(cos_coeff,Uua)),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_145
tff(fact_10353_ATP_Olambda__146,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dx(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_146
tff(fact_10354_ATP_Olambda__147,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_kt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_147
tff(fact_10355_ATP_Olambda__148,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ks(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_148
tff(fact_10356_ATP_Olambda__149,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_wz,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_149
tff(fact_10357_ATP_Olambda__150,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_tq,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= plus_plus_num(l2,l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,plus_plus_num(l3,one)) ) ) ).

% ATP.lambda_150
tff(fact_10358_ATP_Olambda__151,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_sm,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_151
tff(fact_10359_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_nv(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_152
tff(fact_10360_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nx,Uu),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_153
tff(fact_10361_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_nw(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_154
tff(fact_10362_ATP_Olambda__155,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_or,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_155
tff(fact_10363_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_oq,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),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_156
tff(fact_10364_ATP_Olambda__157,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_op,Uu),Uua) = product_Pair_int_int(Uu,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)) ).

% ATP.lambda_157
tff(fact_10365_ATP_Olambda__158,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_pa,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_158
tff(fact_10366_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_oz,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),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_159
tff(fact_10367_ATP_Olambda__160,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_oy,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_160
tff(fact_10368_ATP_Olambda__161,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_on(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_161
tff(fact_10369_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_om(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% ATP.lambda_162
tff(fact_10370_ATP_Olambda__163,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oo(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% ATP.lambda_163
tff(fact_10371_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ix,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_164
tff(fact_10372_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_vn(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),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_165
tff(fact_10373_ATP_Olambda__166,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_wk,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_166
tff(fact_10374_ATP_Olambda__167,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_vl(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_167
tff(fact_10375_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_vi(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_168
tff(fact_10376_ATP_Olambda__169,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rb(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_169
tff(fact_10377_ATP_Olambda__170,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_rd(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_170
tff(fact_10378_ATP_Olambda__171,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rc(Uu),Uua) = minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_171
tff(fact_10379_ATP_Olambda__172,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_re,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_172
tff(fact_10380_ATP_Olambda__173,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_rf(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_173
tff(fact_10381_ATP_Olambda__174,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qj(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_174
tff(fact_10382_ATP_Olambda__175,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_qi(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_175
tff(fact_10383_ATP_Olambda__176,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qk(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_176
tff(fact_10384_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_177
tff(fact_10385_ATP_Olambda__178,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_qm(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_178
tff(fact_10386_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wj,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_179
tff(fact_10387_ATP_Olambda__180,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_fy,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_10388_ATP_Olambda__181,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_fu(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_10389_ATP_Olambda__182,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_fv,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_10390_ATP_Olambda__183,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aTP_Lamm_fw(Uu),Uua)
    <=> aa_num_o(ord_less_eq_num(Uua),Uu) ) ).

% ATP.lambda_183
tff(fact_10391_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fz,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_10392_ATP_Olambda__185,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fx,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_185
tff(fact_10393_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ug,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_186
tff(fact_10394_ATP_Olambda__187,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kz,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_187
tff(fact_10395_ATP_Olambda__188,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_kv(Uu),Uua)
    <=> aa_set_nat_o(ord_less_set_nat(Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_10396_ATP_Olambda__189,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_kw,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_10397_ATP_Olambda__190,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aTP_Lamm_kx(Uu),Uua)
    <=> aa_num_o(ord_less_num(Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_10398_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_jo,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_191
tff(fact_10399_ATP_Olambda__192,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ky,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_192
tff(fact_10400_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_193
tff(fact_10401_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_194
tff(fact_10402_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ui,Uu),Uua) = aa_nat_real(power_power_real(Uua),Uu) ).

% ATP.lambda_195
tff(fact_10403_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_196
tff(fact_10404_ATP_Olambda__197,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_to(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_197
tff(fact_10405_ATP_Olambda__198,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_uk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_198
tff(fact_10406_ATP_Olambda__199,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ah,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_199
tff(fact_10407_ATP_Olambda__200,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_af,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_10408_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ai,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_10409_ATP_Olambda__202,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ag,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_10410_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_sq,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_203
tff(fact_10411_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_204
tff(fact_10412_ATP_Olambda__205,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_vg(Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_205
tff(fact_10413_ATP_Olambda__206,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_a(Uu),Uua)
    <=> member_complex(Uua,Uu) ) ).

% ATP.lambda_206
tff(fact_10414_ATP_Olambda__207,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_207
tff(fact_10415_ATP_Olambda__208,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_208
tff(fact_10416_ATP_Olambda__209,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_209
tff(fact_10417_ATP_Olambda__210,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_210
tff(fact_10418_ATP_Olambda__211,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_211
tff(fact_10419_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mh(Uu),Uua) = aa_nat_real(Uu,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_212
tff(fact_10420_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mg(Uu),Uua) = aa_nat_real(Uu,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_213
tff(fact_10421_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_cz(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_214
tff(fact_10422_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cm(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_215
tff(fact_10423_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_vt(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_216
tff(fact_10424_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_fq(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_217
tff(fact_10425_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_da(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_218
tff(fact_10426_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_db(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_219
tff(fact_10427_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ti(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_220
tff(fact_10428_ATP_Olambda__221,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tf(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_221
tff(fact_10429_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_xh,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_xg(Uu),Uua)) ).

% ATP.lambda_222
tff(fact_10430_ATP_Olambda__223,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sy,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sx(Uu),Uua)) ).

% ATP.lambda_223
tff(fact_10431_ATP_Olambda__224,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sw,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sv(Uu),Uua)) ).

% ATP.lambda_224
tff(fact_10432_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_su,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_st(Uu),Uua)) ).

% ATP.lambda_225
tff(fact_10433_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_ss,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sr(Uu),Uua)) ).

% ATP.lambda_226
tff(fact_10434_ATP_Olambda__227,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sp,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_so(Uu),Uua)) ).

% ATP.lambda_227
tff(fact_10435_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_up(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_uo(Uu),Uua)) ).

% ATP.lambda_228
tff(fact_10436_ATP_Olambda__229,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_mz(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_229
tff(fact_10437_ATP_Olambda__230,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_vy(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_230
tff(fact_10438_ATP_Olambda__231,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_real(aTP_Lamm_ee(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu),Uua))) ).

% ATP.lambda_231
tff(fact_10439_ATP_Olambda__232,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ed(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(power_power_real(Uu),Uua))) ).

% ATP.lambda_232
tff(fact_10440_ATP_Olambda__233,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bd,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(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,Uu),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_233
tff(fact_10441_ATP_Olambda__234,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_eg(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(sin_coeff,Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uu)),Uua))) ).

% ATP.lambda_234
tff(fact_10442_ATP_Olambda__235,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ef(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uu)),Uua))) ).

% ATP.lambda_235
tff(fact_10443_ATP_Olambda__236,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_real(aTP_Lamm_ea(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(sin_coeff,Uua)),aa_nat_complex(power_power_complex(Uu),Uua))) ).

% ATP.lambda_236
tff(fact_10444_ATP_Olambda__237,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_dz(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua))) ).

% ATP.lambda_237
tff(fact_10445_ATP_Olambda__238,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_real(aTP_Lamm_ec(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(cos_coeff,Uua)),aa_nat_complex(power_power_complex(Uu),Uua))) ).

% ATP.lambda_238
tff(fact_10446_ATP_Olambda__239,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_eb(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua))) ).

% ATP.lambda_239
tff(fact_10447_ATP_Olambda__240,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ba(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_240
tff(fact_10448_ATP_Olambda__241,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_bc(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_241
tff(fact_10449_ATP_Olambda__242,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_bb(Uu),Uua) = times_times_rat(minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_242
tff(fact_10450_ATP_Olambda__243,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_bg(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_243
tff(fact_10451_ATP_Olambda__244,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_be(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_244
tff(fact_10452_ATP_Olambda__245,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_bh(Uu),Uua) = times_times_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_245
tff(fact_10453_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bi(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_246
tff(fact_10454_ATP_Olambda__247,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_bf(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_247
tff(fact_10455_ATP_Olambda__248,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_wr(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_248
tff(fact_10456_ATP_Olambda__249,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_vk(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_249
tff(fact_10457_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tz,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_250
tff(fact_10458_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_251
tff(fact_10459_ATP_Olambda__252,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_252
tff(fact_10460_ATP_Olambda__253,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ru,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_253
tff(fact_10461_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_cs(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_254
tff(fact_10462_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ve(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_10463_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_xd(Uu),Uua) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_10464_ATP_Olambda__257,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_fb(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_10465_ATP_Olambda__258,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_complex(aTP_Lamm_fe(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_10466_ATP_Olambda__259,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_fh(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_10467_ATP_Olambda__260,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ff(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_10468_ATP_Olambda__261,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_fg(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_10469_ATP_Olambda__262,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_fd(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_10470_ATP_Olambda__263,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_fc(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_10471_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wv,Uu),Uua) = uminus_uminus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_10472_ATP_Olambda__265,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_pt(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_10473_ATP_Olambda__266,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_pu(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_10474_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_pv(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_10475_ATP_Olambda__268,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_pw(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_10476_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_px(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_10477_ATP_Olambda__270,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_fi(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_10478_ATP_Olambda__271,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_fj(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_10479_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_ow(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_10480_ATP_Olambda__273,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ox(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_10481_ATP_Olambda__274,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_fk(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_274
tff(fact_10482_ATP_Olambda__275,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_fl(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_275
tff(fact_10483_ATP_Olambda__276,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fm(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_276
tff(fact_10484_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_nj(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_10485_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_nn(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_10486_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_nk(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_10487_ATP_Olambda__280,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_nm(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_10488_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_nl(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_281
tff(fact_10489_ATP_Olambda__282,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_wh(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_282
tff(fact_10490_ATP_Olambda__283,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_wg(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_283
tff(fact_10491_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ct(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_284
tff(fact_10492_ATP_Olambda__285,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_jx(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_285
tff(fact_10493_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vq(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_286
tff(fact_10494_ATP_Olambda__287,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_aw(Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = nth_VEBT_VEBT(Uu,I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Uu)) ) ) ).

% ATP.lambda_287
tff(fact_10495_ATP_Olambda__288,axiom,
    ! [Uu: list_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_at(Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_real(nth_real(Uu),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Uu)) ) ) ).

% ATP.lambda_288
tff(fact_10496_ATP_Olambda__289,axiom,
    ! [Uu: list_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_au(Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = nth_list_nat(Uu,I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3023201423986296836st_nat(Uu)) ) ) ).

% ATP.lambda_289
tff(fact_10497_ATP_Olambda__290,axiom,
    ! [Uu: list_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_ax(Uu),(Uua))
    <=> ? [I3: nat] :
          ( ( (Uua)
          <=> aa_nat_o(nth_o(Uu),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Uu)) ) ) ).

% ATP.lambda_290
tff(fact_10498_ATP_Olambda__291,axiom,
    ! [Uu: list_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_av(Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_set_nat(nth_set_nat(Uu),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Uu)) ) ) ).

% ATP.lambda_291
tff(fact_10499_ATP_Olambda__292,axiom,
    ! [Uu: list_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ay(Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_nat(nth_nat(Uu),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Uu)) ) ) ).

% ATP.lambda_292
tff(fact_10500_ATP_Olambda__293,axiom,
    ! [Uu: list_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_az(Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_int(nth_int(Uu),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Uu)) ) ) ).

% ATP.lambda_293
tff(fact_10501_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_dj(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(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(Uu,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_294
tff(fact_10502_ATP_Olambda__295,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mf(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_295
tff(fact_10503_ATP_Olambda__296,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_rj(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),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(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_296
tff(fact_10504_ATP_Olambda__297,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_rg(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),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,Uu))),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_297
tff(fact_10505_ATP_Olambda__298,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rh(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),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,Uu))),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_298
tff(fact_10506_ATP_Olambda__299,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_299
tff(fact_10507_ATP_Olambda__300,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ca(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_300
tff(fact_10508_ATP_Olambda__301,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_301
tff(fact_10509_ATP_Olambda__302,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_cc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_302
tff(fact_10510_ATP_Olambda__303,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_rp(Uu),Uua),Uub) = $ite(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,Uu),Uub))) ).

% ATP.lambda_303
tff(fact_10511_ATP_Olambda__304,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_sk(Uu),Uua),Uub) = $ite(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(Uu),Uub))) ).

% ATP.lambda_304
tff(fact_10512_ATP_Olambda__305,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ro(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_305
tff(fact_10513_ATP_Olambda__306,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_vv(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_306
tff(fact_10514_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nh(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_307
tff(fact_10515_ATP_Olambda__308,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_308
tff(fact_10516_ATP_Olambda__309,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pr(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_309
tff(fact_10517_ATP_Olambda__310,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jw(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jv(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_310
tff(fact_10518_ATP_Olambda__311,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ju(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jt(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_311
tff(fact_10519_ATP_Olambda__312,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_js(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jr(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_312
tff(fact_10520_ATP_Olambda__313,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jq(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jp(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_313
tff(fact_10521_ATP_Olambda__314,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fa(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ez(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_314
tff(fact_10522_ATP_Olambda__315,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ey(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ex(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_315
tff(fact_10523_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_de(Uu),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu,one_one_complex,zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_316
tff(fact_10524_ATP_Olambda__317,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_df(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_317
tff(fact_10525_ATP_Olambda__318,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_dg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu,one_one_int,zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_318
tff(fact_10526_ATP_Olambda__319,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_rn(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_319
tff(fact_10527_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ps(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pr(Uu),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_320
tff(fact_10528_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pp(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_321
tff(fact_10529_ATP_Olambda__322,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ni(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nh(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_322
tff(fact_10530_ATP_Olambda__323,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nf(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ne(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_323
tff(fact_10531_ATP_Olambda__324,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_tw(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_tv(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_324
tff(fact_10532_ATP_Olambda__325,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_tu(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_tt(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_325
tff(fact_10533_ATP_Olambda__326,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_sa(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rz(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_326
tff(fact_10534_ATP_Olambda__327,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ry(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rx(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_327
tff(fact_10535_ATP_Olambda__328,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uq(Uu),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(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_328
tff(fact_10536_ATP_Olambda__329,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_lz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_329
tff(fact_10537_ATP_Olambda__330,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ma(Uu),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(power_power_real(Uu),Uub)) ).

% ATP.lambda_330
tff(fact_10538_ATP_Olambda__331,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_mb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_331
tff(fact_10539_ATP_Olambda__332,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_mc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_332
tff(fact_10540_ATP_Olambda__333,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_md(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_333
tff(fact_10541_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_un(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_334
tff(fact_10542_ATP_Olambda__335,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_me(Uu),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(power_power_real(Uu),Uub)) ).

% ATP.lambda_335
tff(fact_10543_ATP_Olambda__336,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_wo(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_336
tff(fact_10544_ATP_Olambda__337,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_dq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_337
tff(fact_10545_ATP_Olambda__338,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_do(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_338
tff(fact_10546_ATP_Olambda__339,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_wp(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_339
tff(fact_10547_ATP_Olambda__340,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_wn(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_340
tff(fact_10548_ATP_Olambda__341,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_341
tff(fact_10549_ATP_Olambda__342,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_uz(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_342
tff(fact_10550_ATP_Olambda__343,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_si(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_343
tff(fact_10551_ATP_Olambda__344,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_dp(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_344
tff(fact_10552_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_345
tff(fact_10553_ATP_Olambda__346,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ek(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_346
tff(fact_10554_ATP_Olambda__347,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_347
tff(fact_10555_ATP_Olambda__348,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gf(Uu),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,Uu),Uub)) ).

% ATP.lambda_348
tff(fact_10556_ATP_Olambda__349,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_tx(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_349
tff(fact_10557_ATP_Olambda__350,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_sd(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_350
tff(fact_10558_ATP_Olambda__351,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ty(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_351
tff(fact_10559_ATP_Olambda__352,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_sf(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_352
tff(fact_10560_ATP_Olambda__353,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_sg(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_353
tff(fact_10561_ATP_Olambda__354,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_se(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_354
tff(fact_10562_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_uc(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_355
tff(fact_10563_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ud(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_356
tff(fact_10564_ATP_Olambda__357,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ub(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_357
tff(fact_10565_ATP_Olambda__358,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_358
tff(fact_10566_ATP_Olambda__359,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_kn(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_359
tff(fact_10567_ATP_Olambda__360,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_sh(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_360
tff(fact_10568_ATP_Olambda__361,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_co(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_361
tff(fact_10569_ATP_Olambda__362,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_362
tff(fact_10570_ATP_Olambda__363,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_vp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_363
tff(fact_10571_ATP_Olambda__364,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_364
tff(fact_10572_ATP_Olambda__365,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_bq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_365
tff(fact_10573_ATP_Olambda__366,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_366
tff(fact_10574_ATP_Olambda__367,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_go(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_367
tff(fact_10575_ATP_Olambda__368,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_368
tff(fact_10576_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_gr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_369
tff(fact_10577_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sb(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_370
tff(fact_10578_ATP_Olambda__371,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_371
tff(fact_10579_ATP_Olambda__372,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_372
tff(fact_10580_ATP_Olambda__373,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wt,Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_373
tff(fact_10581_ATP_Olambda__374,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_374
tff(fact_10582_ATP_Olambda__375,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_jy(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_375
tff(fact_10583_ATP_Olambda__376,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_kc(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_376
tff(fact_10584_ATP_Olambda__377,axiom,
    ! [Uu: fun_complex_int,Uua: fun_complex_int,Uub: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aTP_Lamm_kg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_complex_int(Uua,Uub)),aa_complex_int(Uu,Uub)) ).

% ATP.lambda_377
tff(fact_10585_ATP_Olambda__378,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_jz(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_378
tff(fact_10586_ATP_Olambda__379,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_kd(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_379
tff(fact_10587_ATP_Olambda__380,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_kh(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu,Uub)) ).

% ATP.lambda_380
tff(fact_10588_ATP_Olambda__381,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ka(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_381
tff(fact_10589_ATP_Olambda__382,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ke(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_382
tff(fact_10590_ATP_Olambda__383,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_kb(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_383
tff(fact_10591_ATP_Olambda__384,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_kf(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ).

% ATP.lambda_384
tff(fact_10592_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_385
tff(fact_10593_ATP_Olambda__386,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wx(Uu),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu,Uub),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_386
tff(fact_10594_ATP_Olambda__387,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ki(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_387
tff(fact_10595_ATP_Olambda__388,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_388
tff(fact_10596_ATP_Olambda__389,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_kk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_389
tff(fact_10597_ATP_Olambda__390,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_km(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_390
tff(fact_10598_ATP_Olambda__391,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_391
tff(fact_10599_ATP_Olambda__392,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_kp(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_392
tff(fact_10600_ATP_Olambda__393,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_393
tff(fact_10601_ATP_Olambda__394,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ww,Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_394
tff(fact_10602_ATP_Olambda__395,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ck(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10603_ATP_Olambda__396,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_cl(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_396
tff(fact_10604_ATP_Olambda__397,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kq(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_10605_ATP_Olambda__398,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_um(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_398
tff(fact_10606_ATP_Olambda__399,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sc(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_399
tff(fact_10607_ATP_Olambda__400,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_400
tff(fact_10608_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pe(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_401
tff(fact_10609_ATP_Olambda__402,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_fp(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_402
tff(fact_10610_ATP_Olambda__403,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_kr(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_403
tff(fact_10611_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_bv(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_404
tff(fact_10612_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_405
tff(fact_10613_ATP_Olambda__406,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_cx(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_406
tff(fact_10614_ATP_Olambda__407,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_407
tff(fact_10615_ATP_Olambda__408,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_uj(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_408
tff(fact_10616_ATP_Olambda__409,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_409
tff(fact_10617_ATP_Olambda__410,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pc(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_410
tff(fact_10618_ATP_Olambda__411,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_pd(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_411
tff(fact_10619_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_wd(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_412
tff(fact_10620_ATP_Olambda__413,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ul(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_413
tff(fact_10621_ATP_Olambda__414,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_cv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_414
tff(fact_10622_ATP_Olambda__415,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mr(Uu),Uua),Uub) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ).

% ATP.lambda_415
tff(fact_10623_ATP_Olambda__416,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ms(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ).

% ATP.lambda_416
tff(fact_10624_ATP_Olambda__417,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_nz(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_417
tff(fact_10625_ATP_Olambda__418,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_418
tff(fact_10626_ATP_Olambda__419,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_oa(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_419
tff(fact_10627_ATP_Olambda__420,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_420
tff(fact_10628_ATP_Olambda__421,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ny(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_421
tff(fact_10629_ATP_Olambda__422,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_lv(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),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,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ly(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_423
tff(fact_10631_ATP_Olambda__424,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_lw(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_424
tff(fact_10632_ATP_Olambda__425,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lx(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_425
tff(fact_10633_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_426
tff(fact_10634_ATP_Olambda__427,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hl(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_427
tff(fact_10635_ATP_Olambda__428,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ho(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_428
tff(fact_10636_ATP_Olambda__429,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_hm(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_429
tff(fact_10637_ATP_Olambda__430,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_hn(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_430
tff(fact_10638_ATP_Olambda__431,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_od(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_431
tff(fact_10639_ATP_Olambda__432,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vx(Uu),Uua),Uub)
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_432
tff(fact_10640_ATP_Olambda__433,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_bx(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_433
tff(fact_10641_ATP_Olambda__434,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_by(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_434
tff(fact_10642_ATP_Olambda__435,axiom,
    ! [Uu: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_pg(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_435
tff(fact_10643_ATP_Olambda__436,axiom,
    ! [Uu: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_pf(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_436
tff(fact_10644_ATP_Olambda__437,axiom,
    ! [Uu: rat,Uua: fun_nat_nat,Uub: nat] : aa_nat_rat(aa_fun2283220954820195200at_rat(aTP_Lamm_ph(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_437
tff(fact_10645_ATP_Olambda__438,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pi(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_438
tff(fact_10646_ATP_Olambda__439,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_pj(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_439
tff(fact_10647_ATP_Olambda__440,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_pk(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_440
tff(fact_10648_ATP_Olambda__441,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qo(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_441
tff(fact_10649_ATP_Olambda__442,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_442
tff(fact_10650_ATP_Olambda__443,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_qp(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_443
tff(fact_10651_ATP_Olambda__444,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_444
tff(fact_10652_ATP_Olambda__445,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_445
tff(fact_10653_ATP_Olambda__446,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lb(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_446
tff(fact_10654_ATP_Olambda__447,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_la(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_447
tff(fact_10655_ATP_Olambda__448,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pm(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_448
tff(fact_10656_ATP_Olambda__449,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_wf(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_449
tff(fact_10657_ATP_Olambda__450,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_cu(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_450
tff(fact_10658_ATP_Olambda__451,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_cg(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_451
tff(fact_10659_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vu(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_452
tff(fact_10660_ATP_Olambda__453,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_na(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_453
tff(fact_10661_ATP_Olambda__454,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pl(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_454
tff(fact_10662_ATP_Olambda__455,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_dc(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_455
tff(fact_10663_ATP_Olambda__456,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_dd(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_456
tff(fact_10664_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_br(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_457
tff(fact_10665_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bp(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub))) ).

% ATP.lambda_458
tff(fact_10666_ATP_Olambda__459,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_real(aa_fun1953249243402654830x_real(aTP_Lamm_qc(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub))) ).

% ATP.lambda_459
tff(fact_10667_ATP_Olambda__460,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_py(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub))) ).

% ATP.lambda_460
tff(fact_10668_ATP_Olambda__461,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_qd(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(Uu,Uub)),aa_real_complex(Uua,Uub))) ).

% ATP.lambda_461
tff(fact_10669_ATP_Olambda__462,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pz(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_462
tff(fact_10670_ATP_Olambda__463,axiom,
    ! [Uu: fun_set_nat_complex,Uua: fun_set_nat_complex,Uub: set_nat] : aa_set_nat_real(aa_fun5767801665285160478t_real(aTP_Lamm_qe(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_set_nat_complex(Uu,Uub)),aa_set_nat_complex(Uua,Uub))) ).

% ATP.lambda_463
tff(fact_10671_ATP_Olambda__464,axiom,
    ! [Uu: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_qa(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(Uu,Uub)),aa_set_nat_real(Uua,Uub))) ).

% ATP.lambda_464
tff(fact_10672_ATP_Olambda__465,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_qh(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub))) ).

% ATP.lambda_465
tff(fact_10673_ATP_Olambda__466,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qg(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_466
tff(fact_10674_ATP_Olambda__467,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_qf(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_int_complex(Uu,Uub)),aa_int_complex(Uua,Uub))) ).

% ATP.lambda_467
tff(fact_10675_ATP_Olambda__468,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_qb(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_468
tff(fact_10676_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vo(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_469
tff(fact_10677_ATP_Olambda__470,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jp(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,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)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_470
tff(fact_10678_ATP_Olambda__471,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jr(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,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)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_471
tff(fact_10679_ATP_Olambda__472,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ex(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,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)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_472
tff(fact_10680_ATP_Olambda__473,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ez(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,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)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_473
tff(fact_10681_ATP_Olambda__474,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jt(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,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)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_complex) ).

% ATP.lambda_474
tff(fact_10682_ATP_Olambda__475,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jv(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,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)),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_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_real) ).

% ATP.lambda_475
tff(fact_10683_ATP_Olambda__476,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_hp(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_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_476
tff(fact_10684_ATP_Olambda__477,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_hx(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,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_477
tff(fact_10685_ATP_Olambda__478,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_hr(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,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_478
tff(fact_10686_ATP_Olambda__479,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_hv(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,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_479
tff(fact_10687_ATP_Olambda__480,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ht(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,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_480
tff(fact_10688_ATP_Olambda__481,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_qw(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,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_481
tff(fact_10689_ATP_Olambda__482,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,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_482
tff(fact_10690_ATP_Olambda__483,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_qy(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,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_483
tff(fact_10691_ATP_Olambda__484,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_qz(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,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_484
tff(fact_10692_ATP_Olambda__485,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ra(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,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_485
tff(fact_10693_ATP_Olambda__486,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_hq(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_486
tff(fact_10694_ATP_Olambda__487,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_hy(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_487
tff(fact_10695_ATP_Olambda__488,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_hs(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_488
tff(fact_10696_ATP_Olambda__489,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_hw(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_489
tff(fact_10697_ATP_Olambda__490,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_hu(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_490
tff(fact_10698_ATP_Olambda__491,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_my(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mx(Uu),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_491
tff(fact_10699_ATP_Olambda__492,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mw(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mv(Uu),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_492
tff(fact_10700_ATP_Olambda__493,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mt(Uu),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_493
tff(fact_10701_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_ij(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          $ite(
            Uuc = zero_zero_nat,
            aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),
            $ite(Uuc = Uu,one_one_Code_integer,zero_z3403309356797280102nteger) )),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_494
tff(fact_10702_ATP_Olambda__495,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ii(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex($ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu,one_one_complex,zero_zero_complex) )),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_495
tff(fact_10703_ATP_Olambda__496,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_il(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(
            Uuc = zero_zero_nat,
            aa_real_real(uminus_uminus_real,Uub),
            $ite(Uuc = Uu,one_one_real,zero_zero_real) )),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_496
tff(fact_10704_ATP_Olambda__497,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ik(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat($ite(
            Uuc = zero_zero_nat,
            uminus_uminus_rat(Uub),
            $ite(Uuc = Uu,one_one_rat,zero_zero_rat) )),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_497
tff(fact_10705_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_ih(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(
            Uuc = zero_zero_nat,
            aa_int_int(uminus_uminus_int,Uub),
            $ite(Uuc = Uu,one_one_int,zero_zero_int) )),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_498
tff(fact_10706_ATP_Olambda__499,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_hf(Uu),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_he(Uu),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_499
tff(fact_10707_ATP_Olambda__500,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_gv(Uu),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_gu(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_500
tff(fact_10708_ATP_Olambda__501,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_gt(Uu),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_gs(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_501
tff(fact_10709_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_gq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_gp(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_502
tff(fact_10710_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_gn(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_gm(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_503
tff(fact_10711_ATP_Olambda__504,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uv(Uu),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(Uu,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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_504
tff(fact_10712_ATP_Olambda__505,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ut(Uu),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(Uu,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(power_power_real(Uub),Uuc)) ).

% ATP.lambda_505
tff(fact_10713_ATP_Olambda__506,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_us(Uu),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(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_506
tff(fact_10714_ATP_Olambda__507,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ur(Uu),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(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_507
tff(fact_10715_ATP_Olambda__508,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lf(Uu),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(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_508
tff(fact_10716_ATP_Olambda__509,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ll(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_509
tff(fact_10717_ATP_Olambda__510,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lh(Uu),Uua),Uub),Uuc) = minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_510
tff(fact_10718_ATP_Olambda__511,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lj(Uu),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,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).

% ATP.lambda_511
tff(fact_10719_ATP_Olambda__512,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_so(Uu),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,Uu),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,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_512
tff(fact_10720_ATP_Olambda__513,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),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_513
tff(fact_10721_ATP_Olambda__514,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_mx(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_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_514
tff(fact_10722_ATP_Olambda__515,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),Uu)),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_515
tff(fact_10723_ATP_Olambda__516,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_hi(Uu),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(Uu),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_516
tff(fact_10724_ATP_Olambda__517,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hk(Uu),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(comm_s7457072308508201937r_real(Uu),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_517
tff(fact_10725_ATP_Olambda__518,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_hj(Uu),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(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_518
tff(fact_10726_ATP_Olambda__519,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_hh(Uu),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(Uu),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_519
tff(fact_10727_ATP_Olambda__520,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ha(Uu),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(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_520
tff(fact_10728_ATP_Olambda__521,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hc(Uu),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(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_521
tff(fact_10729_ATP_Olambda__522,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_hb(Uu),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(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_522
tff(fact_10730_ATP_Olambda__523,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gh(Uu),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,Uu),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_523
tff(fact_10731_ATP_Olambda__524,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_gz(Uu),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(Uu),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_524
tff(fact_10732_ATP_Olambda__525,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_if(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_525
tff(fact_10733_ATP_Olambda__526,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ig(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_526
tff(fact_10734_ATP_Olambda__527,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ln(Uu),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(Uu),Uuc)) ).

% ATP.lambda_527
tff(fact_10735_ATP_Olambda__528,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(Uu),Uuc)) ).

% ATP.lambda_528
tff(fact_10736_ATP_Olambda__529,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_lo(Uu),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(Uu),Uuc)) ).

% ATP.lambda_529
tff(fact_10737_ATP_Olambda__530,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lp(Uu),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(Uu),Uuc)) ).

% ATP.lambda_530
tff(fact_10738_ATP_Olambda__531,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lg(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_531
tff(fact_10739_ATP_Olambda__532,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lm(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_532
tff(fact_10740_ATP_Olambda__533,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_li(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_533
tff(fact_10741_ATP_Olambda__534,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lk(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),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,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_534
tff(fact_10742_ATP_Olambda__535,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_lr(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_535
tff(fact_10743_ATP_Olambda__536,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uu),Uuc)),aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_536
tff(fact_10744_ATP_Olambda__537,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ls(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_537
tff(fact_10745_ATP_Olambda__538,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lt(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_538
tff(fact_10746_ATP_Olambda__539,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gg(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_539
tff(fact_10747_ATP_Olambda__540,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_tv(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_540
tff(fact_10748_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_st(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_541
tff(fact_10749_ATP_Olambda__542,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_tt(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_542
tff(fact_10750_ATP_Olambda__543,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sr(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_543
tff(fact_10751_ATP_Olambda__544,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sv(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_544
tff(fact_10752_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sx(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_545
tff(fact_10753_ATP_Olambda__546,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_xg(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_546
tff(fact_10754_ATP_Olambda__547,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_cq(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_547
tff(fact_10755_ATP_Olambda__548,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_cr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_548
tff(fact_10756_ATP_Olambda__549,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,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_549
tff(fact_10757_ATP_Olambda__550,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ok(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_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_550
tff(fact_10758_ATP_Olambda__551,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_og(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,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_551
tff(fact_10759_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,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_552
tff(fact_10760_ATP_Olambda__553,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_gm(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_553
tff(fact_10761_ATP_Olambda__554,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_gu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_554
tff(fact_10762_ATP_Olambda__555,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_gp(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_555
tff(fact_10763_ATP_Olambda__556,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_he(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_556
tff(fact_10764_ATP_Olambda__557,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_gs(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_557
tff(fact_10765_ATP_Olambda__558,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_nc(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_558
tff(fact_10766_ATP_Olambda__559,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nb(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_559
tff(fact_10767_ATP_Olambda__560,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_pn(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_560
tff(fact_10768_ATP_Olambda__561,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = normalize(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,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_561
tff(fact_10769_ATP_Olambda__562,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_562
tff(fact_10770_ATP_Olambda__563,axiom,
    ! [Uu: 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_mq(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_mp(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_563
tff(fact_10771_ATP_Olambda__564,axiom,
    ! [Uu: 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_mo(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_mn(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_564
tff(fact_10772_ATP_Olambda__565,axiom,
    ! [Uu: 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_mm(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_ml(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_565
tff(fact_10773_ATP_Olambda__566,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_mk(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_mj(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_566
tff(fact_10774_ATP_Olambda__567,axiom,
    ! [Uu: 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_uu(Uu),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_ut(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_567
tff(fact_10775_ATP_Olambda__568,axiom,
    ! [Uu: 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_ol(Uu),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_ok(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_real(power_power_real(Uub),Uud)) ).

% ATP.lambda_568
tff(fact_10776_ATP_Olambda__569,axiom,
    ! [Uu: 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_oj(Uu),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_oi(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_569
tff(fact_10777_ATP_Olambda__570,axiom,
    ! [Uu: 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_oh(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_og(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_570
tff(fact_10778_ATP_Olambda__571,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_of(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oe(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_571
tff(fact_10779_ATP_Olambda__572,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_iu(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_572
tff(fact_10780_ATP_Olambda__573,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_iw(Uu),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(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_573
tff(fact_10781_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_iv(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_574
tff(fact_10782_ATP_Olambda__575,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_hz(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_575
tff(fact_10783_ATP_Olambda__576,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_id(Uu),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(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_576
tff(fact_10784_ATP_Olambda__577,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ib(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_577
tff(fact_10785_ATP_Olambda__578,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ia(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_578
tff(fact_10786_ATP_Olambda__579,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ie(Uu),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(gbinomial_real(aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_579
tff(fact_10787_ATP_Olambda__580,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ic(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(uminus_uminus_rat(Uua)),Uud)),aa_nat_rat(power_power_rat(uminus_uminus_rat(Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_580
tff(fact_10788_ATP_Olambda__581,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_mj(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,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_581
tff(fact_10789_ATP_Olambda__582,axiom,
    ! [Uu: 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_mp(Uu),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(Uu,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(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_582
tff(fact_10790_ATP_Olambda__583,axiom,
    ! [Uu: 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_ml(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,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_583
tff(fact_10791_ATP_Olambda__584,axiom,
    ! [Uu: 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_mn(Uu),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(Uu,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_584
tff(fact_10792_ATP_Olambda__585,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ta(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(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))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I3)),X_12)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X4)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_585
tff(fact_10793_ATP_Olambda__586,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ch(Uu),Uua) = Uu ).

% ATP.lambda_586
tff(fact_10794_ATP_Olambda__587,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ci(Uu),Uua) = Uu ).

% ATP.lambda_587
tff(fact_10795_ATP_Olambda__588,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wu(Uu),Uua) = Uu ).

% ATP.lambda_588
tff(fact_10796_ATP_Olambda__589,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_aj,Uu) = Uu ).

% ATP.lambda_589
tff(fact_10797_ATP_Olambda__590,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ak,Uu) = Uu ).

% ATP.lambda_590
tff(fact_10798_ATP_Olambda__591,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_al,Uu) = Uu ).

% ATP.lambda_591
tff(fact_10799_ATP_Olambda__592,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_am,Uu) = Uu ).

% ATP.lambda_592
tff(fact_10800_ATP_Olambda__593,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_an,Uu) = Uu ).

% ATP.lambda_593
tff(fact_10801_ATP_Olambda__594,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ao,Uu) = zero_zero_complex ).

% ATP.lambda_594
tff(fact_10802_ATP_Olambda__595,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ap,Uu) = zero_zero_real ).

% ATP.lambda_595
tff(fact_10803_ATP_Olambda__596,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_aq,Uu) = zero_zero_rat ).

% ATP.lambda_596
tff(fact_10804_ATP_Olambda__597,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_cd,Uu) = zero_zero_complex ).

% ATP.lambda_597
tff(fact_10805_ATP_Olambda__598,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ce,Uu) = zero_zero_real ).

% ATP.lambda_598
tff(fact_10806_ATP_Olambda__599,axiom,
    ! [Uu: nat] : aa_nat_rat(aTP_Lamm_xc,Uu) = zero_zero_rat ).

% ATP.lambda_599
tff(fact_10807_ATP_Olambda__600,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ar,Uu) = zero_zero_nat ).

% ATP.lambda_600
tff(fact_10808_ATP_Olambda__601,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_cf,Uu) = zero_zero_int ).

% ATP.lambda_601
tff(fact_10809_ATP_Olambda__602,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_as,Uu) = zero_zero_int ).

% ATP.lambda_602
tff(fact_10810_ATP_Olambda__603,axiom,
    ! [Uu: nat] : aa_nat_rat(aTP_Lamm_xf,Uu) = one_one_rat ).

% ATP.lambda_603
tff(fact_10811_ATP_Olambda__604,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ot,Uu) = one_one_nat ).

% ATP.lambda_604
tff(fact_10812_ATP_Olambda__605,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ou,Uu) = one_one_int ).

% ATP.lambda_605
tff(fact_10813_ATP_Olambda__606,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ov,Uu) = one_one_int ).

% ATP.lambda_606
tff(fact_10814_ATP_Olambda__607,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_rr,Uu)
    <=> $false ) ).

% ATP.lambda_607
tff(fact_10815_ATP_Olambda__608,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_rq,Uu)
    <=> $true ) ).

% ATP.lambda_608

% Helper facts (9)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ( (X) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y))
      | ( (X) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X),Y)
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X7: real] : aa_real_o(P,X7) ) ) ).

% Conjectures (10)
tff(conj_0,hypothesis,
    info = none_P5556105721700978146at_nat ).

tff(conj_1,hypothesis,
    zero_zero_nat = deg ).

tff(conj_2,hypothesis,
    ! [X3: vEBT_VEBT] :
      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(treeList))
     => aa_nat_o(vEBT_invar_vebt(X3),na) ) ).

tff(conj_3,hypothesis,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

tff(conj_4,hypothesis,
    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) ).

tff(conj_5,hypothesis,
    m = na ).

tff(conj_6,hypothesis,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

tff(conj_7,hypothesis,
    ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(summary),X_13) ).

tff(conj_8,hypothesis,
    ! [X3: vEBT_VEBT] :
      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(treeList))
     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) ).

tff(conj_9,conjecture,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,n),zero_zero_nat) ).

%------------------------------------------------------------------------------